GeoGebra

Find Angles Between Objects in Space with GeoGebra

You can find the angles between two lines in 3D space, a line and a plane and two planes in 3D space by using GeoGebra. You can also read the entry about drawing lines and planes in 3D space.

GeoGebra Instruction 1

Finding the Angle Between Two Lines

1.: Open Algebra View and 3D Graphics under View in Menu.
2.: In Algebra View, type in the expressions of the two lines that are crossing each other in 3D space. It would be wise to make sure they have a point of intersection, so you know for certain that they will cross each other.
3.: In Algebra View, type

Angle(<line>, <line>)

where you enter the name of each line in each of the <line> fields.

Example 1

Find the angle between the lines $a$ and $b$ with the parametrizations

\begin{array}{l} a : (1, 2, 3) + tFirst, you type the expressions a and b into Algebra View . If you wish to have an illustration of the lines, you can open 3D Graphics under View in Menu . Then use Angle(<line>, <line>) to find the angle between a and b .GeoGebra Instruction 2 Finding the Angle Between a Line and a Plane 1. Open Algebra View and 3D Graphics under View in Menu . 2. In Algebra View type         in         the         expressions         of         your         line         and         plane         that         are         crossing         each         other.         It         would         be         wise         to         make         sure         they         have         a         point         of         intersection,         so         that         they         will         cross         each         other. 3. In Algebra View type in Angle(<line>, <plane>) where you type in the name of the line in the <line> field and the name of the         plane in the <plane> field.Example 2 Find the angle between the line a and the plane β, where \begin{array}{l} a : (1, 2, 3) + t [2, 1, 4] \\ β : y + z - 5 = 0 \end{array} You type in the expressions of a and β in Algebra View . If you wish to have an illustration, you can open 3D Graphics under View in Menu . Then use Angle(<Line>, <Plane>) to find the angle between the line a and the plane β .GeoGebra Instruction 3 Finding the Angle Between Two Planes 1. Open Algebra View and 3D Graphics under View in Menu . 2. In Algebra View type         in         the         expressions         for         two         intersecting         place.         Make         sure         they         actually         intersect. 3. In Algebra View type         in Angle(<plane>, <plane>) where         you         type         in         the         names         of         the         planes         in         the <plane> fields.Example 3 Find the angle between the plane α and the plane β given by \begin{array}{l} α : - 13 x + 14 y + 7 z = - 36 = 0 \\ β : y + z - 5 = 0 \end{array} You type in the expressions of α and β in Algebra View . If you wish to have an illustration, you can open 3D Graphics under View in Menu . Then use Angle(<plane>, <plane>) to find the angle between the planes α and β .Want to know more? Sign Up It's free!DoneContinue32How to Find Distance in GeoGebra33How to Make a Sphere in GeoGebra34How to Find Intersection in GeoGebra35Completing the Square Using GeoGebra36How to Solve an Equation in GeoGebra37How to Solve Systems of Equations in GeoGebra38Solve Multivariate Systems of Equations in GeoGebra39How to Solve Inequalities in GeoGebra40How to Compute Logarithms in GeoGebra41Figures, Perimeter (Circumference) and Area in GeoGebra42How to Draw Loci with GeoGebra43Check If Two Figures are Congruent Using GeoGebra44How to Find Circumcircle and Circumcenter with GeoGebra45How to Find Incircle and Incenter Using GeoGebra46How to Find Centroid and Medians with GeoGebra47How to Find Orthocenter and Altitudes with GeoGebra48How to Find the Euler Line of a Triangle in GeoGebra49How to Make a Circle in GeoGebra50How to Graph Circles of Apollonius in GeoGebra51How to Study the Power of a Point Theorem with GeoGebra52How to Graph an Ellipse in GeoGebra53Translation of Figures in GeoGebra54Constructing Parallels in GeoGebra55Constructing a Parallel through a Point in GeoGebra56How to Do a Reflection in GeoGebra57Constructing the Tangents to a Circle in GeoGebra58How to Create an Angle in GeoGebra59How to Construct Supplementary Angles in GeoGebra60How to Check if Two Angles are Equal with GeoGebra61Inscribed and Central Angles in GeoGebra62How to Rotate a Figure in GeoGebra63Constructing the Normal Through a Point in GeoGebra64How to Construct a Perpendicular Bisector in GeoGebra65How to Construct an Angle Bisector in GeoGebra66How to Make a Polyhedron in GeoGebra67How to Draw Prisms in GeoGebra68How to Draw Cylinders in GeoGebra69How to Make Pyramids in GeoGebra70How to Make a Cone in GeoGebra71How to Make a Sphere in GeoGebra72Surface Area and Volume of Solid Figures in GeoGebra73How to Make a Unit Circle in GeoGebra74How to Use Trigonometric Functions in GeoGebra75Calculate with the Law of Sines and Cosines in GeoGebra76How to Solve Trigonometric Equations in GeoGebra77How to Graph a Trigonometric Function in GeoGebra78Transforming into a Harmonic Oscillator in GeoGebra79Perspective Drawing in GeoGebra80How to Make a Frequency Table with GeoGebra81How to Make a Bar Chart with GeoGebra82How to Make a Pie Chart with GeoGebra83How to Make a Line Chart with GeoGebra84How to Make a Box Plot with GeoGebra85How to Make a Histogram with GeoGebra86How to Calculate Central Tendencies with GeoGebra87Relative Frequency and Law of Large Numbers in GeoGebra88Compute Permutations and Combinations in GeoGebra89How to Use GeoGebra as Binomial Distribution Calculator90Hypergeometric Distribution Calculator in GeoGebra91How to Use GeoGebra as a Normal Distribution Calculator92Hypothesis Testing of Binomial Distribution in GeoGebra93Hypothesis Testing with Normal Distribution in GeoGebra94How to Graph a Function in GeoGebra95How to Find Domain and Range of a Function in GeoGebra96How to Find Intersection with the Axes in GeoGebra97How to Find Intersection Between Graphs in GeoGebra98How to Find the Zeros of a Function in GeoGebra99How to Show Asymptotes in GeoGebra100How to Draw a Tangent to a Curve in GeoGebra101How to Animate Instantaneous Rate of Change in GeoGebra102How to Calculate Limits of Functions Using GeoGebra103How to Find the Derivative of a Function in GeoGebra104How to Find Stationary Points in GeoGebra105How to Find Inflection Points in GeoGebra106How to Use Integral in GeoGebra107How to Find the Area Under a Graph Using GeoGebra108How to Find the Area Between Graphs Using GeoGebra109How to Find Solid of Revolution in GeoGebra110Integral Curves and Direction Fields in GeoGebra111How to Write a Differential Equation in GeoGebra112How to Do Regression in GeoGebra113How to Solve Linear Optimization Problems with GeoGebraHome Curriculum Stats LeagueAI Ask the OracleAI How can I help you? BetaAbout Us About House of Math Employees Career Media Lectures Blog Contact support@houseofmath.com Booking of private tutoring +47 22 150 300 Address Bygdøy allé 23
            0262 Oslo
            Norway FAQ Links WeTakeAction Math Magic for Ukraine Tutoring Bootcamps Junior Math Pen & Paper exercises Encyclopedia Book a Tutor Language/Curriculum Terms Privacy Download our app{"props":{"pageProps":{"campaign":false,"pageData":{"activities":[],"chest":{},"content":[],"encyclopediaArticle":{"breadcrumbs":[{"_key":"bcedd644-0bec-506b-86c8-f5dfe20c2b99","h1":"GeoGebra","path":"/geogebra"},{"_key":"ef61a895-ac2e-50db-8973-568345df9c6a","h1":"Numbers and Quantities","path":"/geogebra/numbers-and-quantities"},{"_key":"e0c1b714-a308-5187-9ae8-da563b5bd966","h1":"Vectors","path":"/geogebra/numbers-and-quantities/vectors"},{"_key":"225d9721-9be2-52ce-bdde-c3e13b3fcef0","h1":"Three Dimensions","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions"},{"_key":"ff5fd763-7e2e-512a-9ec2-5fe3a9052078","h1":"Find Angles Between Objects in Space with GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/find-angles-between-objects-in-space-with-geogebra"}],"cssUrl":"/0e751716-e4cf-4658-ac87-c97d9167170e-en-2.css","h1":"Find Angles Between Objects in Space with GeoGebra","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     You can find the angles between two \u003ca href=\"/encyclopedia/numbers-and-quantities/vectors/three-dimensions/parametrization/parameterization-of-lines-in-space\"\u003elines\u003c/a\u003e in 3D space, a line and a \u003ca href=\"run:encyclopedia/Planes\"\u003eplane\u003c/a\u003e and two planes in 3D space by using \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e. You can also read the entry about \u003ca href=\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-make-lines-and-planes-in-geogebra\"\u003edrawing lines and planes in 3D space\u003c/a type=\"punct\"\u003e. \u003c/p\u003e      \u003cdiv class=\"tcolorbox foogeogebra\" id=\"tcolobox-563\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e Instruction 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"geogebra-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eFinding\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ethe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eAngle\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eBetween\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eTwo\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eLines\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Open         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     and         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e3D\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGraphics\u003c/span\u003e\u003c/code\u003e     under         \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     in         \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eMenu\u003c/span\u003e\u003c/code type=\"punct\"\u003e.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         In         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e,         type         in         the         expressions         of         the         two         lines         that         are         crossing         each         other         in         3D         space.         It         would         be         wise         to         make         sure         they         have         a         point         of         intersection,         so         you         know         for         certain         that         they         \u003cspan  class=\"ec-lmri-12x-x-144\"\u003ewill\u003c/span\u003e        cross         each         other.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eIn \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e, type         \u003cdiv class=\"flushleft\"  \u003e \u003cp class=\"noindent\" \u003e   \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAngle(\u0026#x003C;line\u0026#x003E;, \u0026#x003C;line\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e     \u003cp class=\"noindent\" \u003ewhere you enter the name of each line in each of the \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e\u0026#x003C;line\u0026#x003E;\u003c/span\u003e\u003c/code\u003e fields. \u003c/p\u003e\u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                         \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-564\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eFind the angle between the lines \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eand \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ewith the parametrizations\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" mathvariant=\"bold-italic\"\u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                              \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmi  \u003et\u003c/mi\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e                              \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmi  \u003et\u003c/mi\u003e\u003cmspace width=\"-0.17em\" class=\"negthinspace\"/\u003e \u003cmrow\u003e\u003cmo fence=\"true\" form=\"prefix\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\"true\" form=\"postfix\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eFirst, you type the expressions \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math\u003e into \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e. If you wish to have an illustration of the lines, you can open \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e3D Graphics\u003c/span\u003e\u003c/code\u003e under \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e in \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eMenu\u003c/span\u003e\u003c/code\u003e. Then use \u003c/p\u003e \u003cdiv class=\"flushleft\"  \u003e \u003cp class=\"noindent\" \u003e   \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAngle(\u0026#x003C;line\u0026#x003E;, \u0026#x003C;line\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e \u003cp class=\"noindent\" \u003eto find the angle between \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e      \u003cdiv class=\"tcolorbox foogeogebra\" id=\"tcolobox-565\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e Instruction 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"geogebra-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eFinding\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ethe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eAngle\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eBetween\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ea\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eLine\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eand\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ea\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ePlane\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Open         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     and         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e3D\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGraphics\u003c/span\u003e\u003c/code\u003e     under         \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     in         \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eMenu\u003c/span\u003e\u003c/code type=\"punct\"\u003e.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         In         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     type         in         the         expressions         of         your         line         and         plane         that         are         crossing         each         other.         It         would         be         wise         to         make         sure         they         have         a         point         of         intersection,         so         that         they         will         cross         each         other.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003eIn \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e type in         \u003cdiv class=\"flushleft\"  \u003e \u003cp class=\"noindent\" \u003e   \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAngle(\u0026#x003C;line\u0026#x003E;, \u0026#x003C;plane\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e     \u003cp class=\"noindent\" \u003ewhere you type in the name of the line in the \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e\u0026#x003C;line\u0026#x003E;\u003c/span\u003e\u003c/code\u003e field and the name of the         plane in the \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e\u0026#x003C;plane\u0026#x003E;\u003c/span\u003e\u003c/code\u003e field. \u003c/p\u003e\u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                         \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-566\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eFind the angle between the line \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eand the plane \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003e,\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003ewhere\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" mathvariant=\"bold-italic\"\u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                               \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e)\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmi  \u003et\u003c/mi\u003e\u003cmo  class=\"MathClass-open\" stretchy=\"false\"\u003e[\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" stretchy=\"false\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-close\" stretchy=\"false\"\u003e]\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eYou type in the expressions of \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math\u003e in \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e. If you wish to have an illustration, you can open \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e3D Graphics\u003c/span\u003e\u003c/code\u003e under \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e in \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eMenu\u003c/span\u003e\u003c/code\u003e. Then use  \u003c/p\u003e \u003cdiv class=\"flushleft\"  \u003e \u003cp class=\"noindent\" \u003e   \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAngle(\u0026#x003C;Line\u0026#x003E;, \u0026#x003C;Plane\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e \u003cp class=\"noindent\" \u003eto find the angle between the line \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and the plane \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e      \u003cdiv class=\"tcolorbox foogeogebra\" id=\"tcolobox-567\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e Instruction 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003ch3 class=\"geogebra-container-title\"\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eFinding\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ethe\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eAngle\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eBetween\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003eTwo\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-172\"\u003ePlanes\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\"enumerate-enumitem\"\u003e\u003cdt class=\"enumerate-enumitem\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         Open         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     and         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e3D\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eGraphics\u003c/span\u003e\u003c/code\u003e     under         \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     in         \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eMenu\u003c/span\u003e\u003c/code type=\"punct\"\u003e.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         In         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     type         in         the         expressions         for         two         intersecting         place.         Make         sure         they         actually         intersect.         \u003c/dd\u003e\u003cdt class=\"enumerate-enumitem\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\"enumerate-enumitem\"\u003e         In         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e     type         in         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAngle(\u0026#x003C;plane\u0026#x003E;,\u003c/span\u003e     \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e\u0026#x003C;plane\u0026#x003E;)\u003c/span\u003e\u003c/code\u003e     where         you         type         in         the         names         of         the         planes         in         the         \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e\u0026#x003C;plane\u0026#x003E;\u003c/span\u003e\u003c/code\u003e     fields.         \u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                         \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-568\"\u003e      \u003cdiv class=\"tcolorbox-title\"\u003e \u003cp class=\"indent\" \u003e     Example 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003e\u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eFind the angle between the plane\u003c/span\u003e \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eα\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eand the\u003c/span\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003eplane \u003c/span\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\"ec-lmbx-12x-x-144\"\u003egiven by\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" mathvariant=\"bold-italic\"\u003e\u003cmtable  displaystyle=\"true\" columnalign=\"left\" class=\"align-star\"\u003e                        \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eα\u003c/mi\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmn\u003e7\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e                        \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e \u003cmo  class=\"MathClass-punc\" stretchy=\"false\" type=\"divide\"\u003e:\u003c/mo\u003e \u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e+\u003c/mo\u003e \u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\"MathClass-bin\" stretchy=\"false\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\"MathClass-rel\" stretchy=\"false\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\"lowerbox\"\u003e                                                                                                                                                                                                                                           \u003cp class=\"noindent\" \u003eYou type in the expressions of \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eα\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math\u003e in \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e. If you wish to have an illustration, you can open \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003e3D Graphics\u003c/span\u003e\u003c/code\u003e under \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eView\u003c/span\u003e\u003c/code\u003e in \u003cspan class=\"fbox\"\u003e\u003cimg src=\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\" alt=\"GeoGebra icon\" width=\"19\" height=\"19\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eMenu\u003c/span\u003e\u003c/code\u003e. Then use  \u003c/p\u003e \u003cdiv class=\"flushleft\"  \u003e \u003cp class=\"noindent\" \u003e   \u003cspan  class=\"ec-lmtt-12x-x-144\"\u003eAngle(\u0026#x003C;plane\u0026#x003E;, \u0026#x003C;plane\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e \u003cp class=\"noindent\" \u003eto find the angle between the planes \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eα\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e \u003cp class=\"indent\" \u003e     "},"image":null,"isRootPath":false,"key":"content/en/software/02 GeoGebra/01 Numbers and Quantities/03 Vectors/02 Three Dimensions/04AnglesSpaceGG.tex","language":"en","locale":"en","metaDescription":"Learn how to use GeoGebra to find the angle between two lines, a line and a plane or two planes, in 3D space. Discover the flexibility of this program.","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/find-angles-between-objects-in-space-with-geogebra","previewUrl":"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy8wNGFuZ2xlc3NwYWNlZ2cucG5nIiwiYnVja2V0IjoiaG9tLWZyb250ZW5kLWFzc2V0cyIsInZlcnNpb24iOjN9","sharedLanguageKey":null,"siblings":[{"h1":"How to Compute Vector Products with GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-compute-vector-products-with-geogebra"},{"h1":"How to Make Lines and Planes in GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-make-lines-and-planes-in-geogebra"},{"h1":"Find Angles Between Objects in Space with GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/find-angles-between-objects-in-space-with-geogebra"},{"h1":"How to Find Distance in GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-find-distance-in-geogebra"},{"h1":"How to Make a Sphere in GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-make-a-sphere-in-geogebra"},{"h1":"How to Find Intersection in GeoGebra","path":"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-find-intersection-in-geogebra"}],"taskReference":null,"title":"Find Angles Between Objects in Space with GeoGebra","type":"geogebra"},"help":{"encyclopediaPath":"","drillPath":""},"sectionName":"Find Angles Between Objects in Space with GeoGebra","usefulWords":[],"valid":true,"metaDescription":"Learn how to use GeoGebra to find the angle between two lines, a line and a plane or two planes, in 3D space. Discover the flexibility of this program.","noindex":false},"_nextI18Next":{"initialI18nStore":{"en":{"imageAlt":{"icons":{"houseOfMath":"Colorful House of Math logo","homShortLogo3D":"White H House of Math logo on multicolored square","chat":"Purple speech balloon chat icon","dashboard":"Purple dashboard icon represented as a browser window","exploreMobile":"Magnifying glass exploring planet Earth","menu":"Menu icon","logout":"Open door with a purple arrow for logging out","settings":"Purple gear icon for settings","search":"Magnifying glass for search function","language":"Language","norwegian":"Norsk","english":"English","ukrainian":"Українська","explore":"Magnifying glass closely exploring planet Earth","comingSoon":"Yellow hanging sign that reads Coming Soon","playButton":"Play button icon","activitySetIcon":"Activity set icon","adminPanel":"Purple control panel","arrowLeft":"White arrow pointing to the Left","arrowRight":"White arrow pointing to the right","products":{"mathTopics":"All math","bootcamp":"Purple math kettlebell and gray dummbell","games":"Purple plus sign with video game controls on it","digitalTools":"Laptop and desklamp on purple desk with books and calculator","encyclopedia":"Encyclopedia volume with math symbols on pages","mathVault":"Purple bank vault with round, golden door","mathEssentials":"Black graduate's cap with a purple tassel","tutorOnDemand":"Bespectacled math expert wearing a purple tie","studyTips":"Yellow math study tips lightbulb","exams":"Purple examination book with right and wrong answer"},"gemstoneOutlineWhite":"Outline of gemstone in white","facebook":"Facebook icon","google":"Google icon","mail":"Email icon","tiktok":"TikTok icon","stars":"Stars","reviewFrom":"Review from {{ name }}","quotationMark":"Quotation Mark","home":"Home icon","leagues":"Leaderboard icon","space":"Dashboard icon","icon":"icon"},"emralGem":"Sparkling green emerald","rubyGem":"Shining red ruby","diamondGem":"Sparkling blue-white diamond","spriteSheet":"Treasure chest","loginBeanstalk":"Sing-In Sign-Up Bean Stalk","streakFlame":"Streak Flame","homBeanstalk":"House of Math Bean Stalk","ribbon":"Ribbon","badge":"Badge","badgeShield":"Badge Shield","multMasterLogo":"Multiplication - The Championship","numberLineIncreasing":"Number line increasing","rightArrow":"Right arrow","timesIcon":"Times icon","purpleUpwardsArrow":"Purple upwards arrow","avatarFrame":"Avatar frame"},"curriculum":{"first":"1st","second":"2nd","third":"3rd","gradeCurriculum":"{{gradeText}} grade math","exploreMathVault":"Explore the mysterious math vault and open all the chests with multiple choice activities. Embark on a mission to \u003cbr/\u003e\u003cstrong\u003emaster the fundamentals of math!\u003c/strong\u003e","enterVault":"Enter the vault","add":"Set as current curriculum","remove":"Remove as current curriculum","viewOtherGrades":"View other grades","getStarted":"To get started","gotoPrevGrade":"Go to {{prevGradeLabel}}","gotoNextGrade":"Go to {{nextGradeLabel}}","addedSuccessfully":"Set successfully","removedSuccessfully":"Removed successfully","overview":"Overview","toHomeButtonText":"To Home","dropdownTitle":"Curriculum","norwegianAvailability":"The Norwegian curriculum is only available in Norwegian","curriculumChangeWarning":"\u003cbold\u003eNote:\u003c/bold\u003e Changing this setting may switch the platform language, adjust some math content, and modify your selected curriculum.","changeCurriculum":"Only change curriculum","changeCurriculumAndLanguage":"Change both curriculum and platform language","goal":"Goal {{code}}","viewGrade":"View curriculum","ChangeGrade":"Change your grade","confirm":"Confirm selection","mathematicsMasteryCertificate":"Mathematics Mastery Certificate","awardedTo":"Awarded to","correctness":"Correctness","date":"Date","signed":"Signed","mastered":"You have mastered","clickToViewCertificate":"Click to view your certificate","content":"Content","mappingQuiz":{"mappingQuiz":"Mapping Quiz","stepOneBuildMasteryTrack":"Step 1: Build Mastery Track","mapBaseKnowledge":"Map your base knowledge to create a optimized mastery track","createMasteryTrack":"Create mastery track","tellMeWhyCreateMasterTrack":"Tell me why I should create Master Track","noMathGaps":"Great job! It looks like you don’t have any math gaps right now. Want to map again and double-check?","takeMappingQuiz":"Take Mapping quiz","tryAgain":"Try again","showOnlyCurriculum":"Show only curriculum","completeTopics":"Complete these topics to fill your gaps","generateNewPath":"Generate new path","prerequisites":"Prerequisites","missingCoreSkills":"Missing core skills","missingSingleCoreSkill":"Missing core skill","getStarted":"Start collecting pieces","doMath":"Do math","completion":{"withMultiplePrerequisitesMessage":"You’re missing {{totalPrerequisites}} \u003cgreen\u003ecore skills\u003c/green\u003e before you can fully start mastering \u003citalic\u003e{{subject}}\u003c/italic\u003e with eaze.\nLet’s start building math skills","withSinglePrerequisiteMessage":"You’re missing 1 \u003cgreen\u003ecore skill\u003c/green\u003e before you can fully start mastering \u003citalic\u003e{{subject}}\u003c/italic\u003e with eaze.\nLet’s start building math skills","noPrerequisitesMessage":"You’re not missing any \u003cgreen\u003ecore skills\u003c/green\u003e – everything looks to be in place!"},"generatingPathToMasterPath":"Generating your path to master math","takeMQ":"Take Mapping quiz","mappingYourKnowledge":"Start by mapping your knowledge. Take the Mapping Quiz","completeSubTopics":"Complete these sub-topics to master your curriculum","topicsIn":"levels in","showMorePrerequisites":"Show more prerequisites","showLess":"Show less","pageTitleNoMQ":"For Mastery Learning Track take the Mapping Quiz","masteryLearningTrack":"Mastery Learning Track","masteryCertificate":"Mastery Certificate","stepOne":"Step 1","stepTwo":"Step 2","noCertificates":"Complete topics in your curriculum to earn mastery certificates","latestMateryCertificates":"Latest Mastery Certificates","seeAll":"See all","goToCurriculum":"Go to curriculum","empty":"Empty"},"awarded":"Awarded","grade":"{{grade}}th Grade","mathLearner":"Math learner","onboarding":{"greatChoice":"Great Choice","greatChoiceDescription":"Great choice!\u003cbr/\u003eYou are about to master\u003cbr/\u003e\u003citalic\u003e{{subject}}!\u003c/italic\u003e","coreSkillsDescription":"Learning math requires you to build knowledge stone by stone. If building blocks are missing, the structure will collapse.","learning":"Learning","title":"How do you want to start learning \u003citalic\u003e{{title}}\u003c/italic\u003e ?","burnZone":"Burn zone","whatIsBurnZone":"What is the Burn Zone?","burnZoneDescription":"\u003ctitle\u003eThis is how you become good at \u003citalic\u003e{{subject}}\u003c/italic\u003e\u003c/title\u003e\u003cbr/\u003e\u003cbr/\u003eLacking the core skills causes you to fail at math. When the \u003cinline\u003e\u003cCoreSkills /\u003e\u003cpuzzleImage /\u003e\u003c/inline\u003e are not in place, learning becomes overwhelming, and you end up in the Burn Zone.","burnZoneModalDescription":"The Burn Zone is when a student is pushed too far and experiences frustration, stress, or burnout.\u003cbr/\u003e\u003cbr/\u003eYou don’t want to be here.\u003cbr/\u003e \u003cbr/\u003eWe will show you how to get out of the burn zone.","tellMeMore":"Tell me more","recommended":"Recommended","buildMasteryTrack":"Mastery Track","personalLearningPath":"your personal learning path","buildMasteryTrackDescription":"This will create a personal learning path, ensuring that you are not \u003cbold\u003emissing any core pieces.\u003c/bold\u003e","or":"or","jumpBtnText":"Without help","jumpBtnDescription":"You can start now, but if you're missing \u003cem\u003ecore pieces,\u003c/em\u003e some problems may be much harder.","whyMQ":"Why pick mastery Track","coreSkills":"Core skills","coreSkillsBurnPage":"Core skills","knowledge":"Knowledge","smartChoice":"Smart choice!","unlockPotential":"Let's unlock your full potential","knowledgeDialog1":"We will now map your \u003cbr/\u003e\u003copacity\u003e\u003cel\u003eBase knowledge\u003c/el\u003e\u003c/opacity\u003e to master \u003cbr/\u003e \u003citalic\u003e{{title}}\u003c/italic\u003e","knowledgeDialog2":"Once we have your \u003copacity\u003e\u003cel\u003eBase knowledge\u003c/el\u003e\u003c/opacity\u003e,\u003cbr/\u003ewe will know where you should start \u003cbr/\u003e\u003copacity\u003e\u003cem\u003ebuilding your competency\u003c/em\u003e\u003c/opacity\u003e","knowledgeDialog3":"So you don't end up in your \u003copacity\u003e\u003cbl\u003eBurn zone\u003c/bl\u003e\u003c/opacity\u003e","trackDecision":"Track decision","trackDecisionTitle":"Are you sure you want to skip the Mastery Track?","trackDecisionDescription":"Jumping straight in will be harder if you're missing \u003cinline\u003e\u003cCoreSkills /\u003e\u003cpuzzleImage /\u003e\u003c/inline\u003e","trackBtn1":"Yes, I know I'm not missing any core skills","trackBtn2":"No, build Mastery Track"}},"footer":{"about":"About Us","aboutHoM":"About House of Math","aboutTeam":"Employees","blog":"Blog","bootcamps":"Bootcamps","career":"Career","contact":"Contact","cookies":"Cookies","encyclopedia":"Encyclopedia","lectures":"Lectures","links":"Links","mathEssentials":"Math Essentials","mathVault":"Junior Math","mathMagic":"Math Magic for Ukraine","media":"Media","norway":"Norway","privacy":"Privacy","pricing":"Pricing","terms":"Terms","tutoring":"Tutoring","weTakeAction":"WeTakeAction","upgrade":"Upgrade Now","bookingOfPrivateTutoring":"Booking of private tutoring","address":"Address","bookTutor":"Book a Tutor","penNPaperExercise":"Pen \u0026 Paper exercises","footerCurriculumTitle":"Language/Curriculum","downloadApp":"Download our app"},"generics":{"active":"Active","add":"Add","addComment":"Add comment","admin":"Admin","adminPanel":"Admin Panel","administration":"Administration","afternoon":"Afternoon","apply":"Apply","appointments":"Appointments","avatar":"User avatar","ariaLabelButtons":{"adjustSliderToMatchFraction":"Adjust slider to match fraction","backToBootcamp":"Back to bootcamp","checkBox":"Check box","clear":"Clear","closeWindow":"Close window","closeChatWindow":"Close chat window","collapseSessionDetails":"Collapse session details","copyToClipboard":"Copy to clipboard","decimalMark":"Decimal mark","expandSessionDetails":"Expand session details","expandDropdownMenu":"Expand dropdown menu","info":"Info","insertArea":"Insert area","insertDecimalValue":"Insert decimal value","insertCarriedNumber":"Insert carried number","insertCoefficientValue":"Insert coefficient value","insertCommonFactor":"Insert common factor","insertDenominator":"Insert denominator","insertFactor":"Insert factor","insertLatticeSum":"Insert lattice sum","insertNumberOfEdges":"Insert number of edges","insertNumerator":"Insert numerator","insertOne":"Insert ones","insertPrimeFactor":"Insert prime factor","insertProduct":"Insert product","insertResult":"Insert result","insertSimplifiedExpression":"Insert simplified expression","insertSum":"Insert sum","insertTen":"Insert tens","insertTerm":"Insert term","insertValue":"Insert value","reset":"Reset","reload":"Reload","removeFromSearchHistory":"Remove from search history","submit":"Submit"},"back":"Back","backButton":"Back Button","backTo":"Back to {{ pageTitle }}","bean":"Beanstalk","blog":"Blog","book":"Book","bookAgain":"Book again","cancel":"Cancel","chat":"Chat","checkAnswer":"Check answer","city":"City","claim":"Claim","clickHere":"Click here","close":"Close","code":"Code","comingSoon":"Coming soon","condense":"Condense","confirm":"Confirm","continue":"Continue","copied":"Copied","correct":"Correct","countryCode":"Country code flag","incorrect":"Incorrect","currency":"Currency","date":"Date","daysShort":{"fri":"Fr","mon":"Mo","sat":"Sa","sun":"Su","thu":"Th","tue":"Tu","wed":"We"},"digital":"Digital","edit":"Edit","email":"E-mail","erase":"Erase","expand":"Expand icon","reset":"Reset","evening":"Evening","excellent":"Excellent","facebookEmail":"Facebook e-mail","finished":"Finished","from":"From","funFact":"Fun Fact","getFullAccessNow":"Get full access now","go":"Go","goTo":"Go to","goToHomebase":"Space","good":"Good","gradientColorCircle":"Gradient color circle","hour":"Hour","hours":"Hours","hoursAbbr":"H","icon":"Icon","inactive":"Inactive","language":"Language","lastSession":"Last session","later":"Later","leafAlt":"Green leaf","log":"Log","logIn":"Log in","goToDashboard":"Go to dashboard","mathMaster":"Math Master","subscribe":"Subscribe","tutor":"Tutor","tutorChatPanel":"Tutor Chat","tutorOnDemand":"Tutor-on-Demand","mentorPanel":"Tutor Panel","tutors":"Tutors","message":"Message","minutes":"Minutes","monthsLong":{"apr":"April","aug":"August","dec":"December","feb":"February","jan":"January","jul":"July","jun":"June","mar":"March","may":"May","nov":"November","oct":"October","sep":"September"},"moreInformation":"More information","morning":"Morning","name":"Name","neutral":"Neutral","next":"Next","nextQuestion":"Next question","nextSession":"Next session","no":"No","noTutors":"No tutors","observe":"Observe","ok":"Ok","openSession":"Open session","or":"Or","orderDate":"Order date","orderId":"Order ID","person":"Person","phoneNumber":"Phone number","play":"Play","playAlt":"Play","poor":"Poor","previous":"Previous","price":"Price","product":"Product","purchasedAt":"Purchased at","purpleCheckIcon":"Purple check icon","scheduledAt":"Scheduled at","readMore":"Read more","receipt":"Receipt","resetButton":"Reset button","role":"Role","save":"Save","seeMore":"See more","send":"Send","sessionStatus":"Session status","showAll":"Show all","showingAllFromAndIncluding":"Showing all from and including {{ date }}","signUp":"Sign up","signUpFree":"Sign up for free","skill":{"average":"Average","strong":"Strong","veryStrong":"Very strong","aBitWeak":"A bit weak","weak":"Weak"},"skip":"Skip","startTime":"Start time","status":"Status","stay":"Stay","student":"Student","studentPanel":"Student Panel","studentPanelToolTipContent":"Student Panel with a Full Overview of Your Activities. When you book a tutor or participate in a video chat with a tutor, you get access to your private student panel.","students":"Students","subTotal":"Product price","subjects":{"biology":"Biology","chemistry":"Chemistry","economy":"Economy","finance":"Finance","math":"Math","other-subjects":"Other subjects","physics":"Physics","statistics":"Statistics"},"tellMeHow":"Tell me how","terrible":"Terrible","thanks":"Thanks","time":"Time","to":"To","total":"Total","travelExpense":"Travel expense","tryAgain":"Try again","tutoring":"Tutoring","update":"Update","updateSuccess":"Update successful","user":"User","videoChat":"Video Chat","videoChatTutorPanel":"Video Chat Tutor Panel","week":"Week","yes":"Yes","emptyGem":"Empty gem","filledGem":"Filled gem","studentHistory":"Student History","studentSettings":"Student Settings","last_item":"add above this","userId":"User ID","bookTutor":"Book Tutor","popUpCouponUpdated":"\u003cred\u003eImportant Update:\u003c/red\u003e Beginning on November 16th, House of Math will introduce a subscription version. To express our gratitude for your continued support and patience, we want to offer you the first month at an exclusive rate of just 5 NOK!","norwegianCouponUpdated":"Get your \u003csemiBold\u003efirst month\u003c/semiBold\u003e for just \u003csemiBold\u003e5 Kr\u003c/semiBold\u003e as a huge thank you for your amazing support!\u003csuccessImage /\u003e \nSubscribe now!","bigNewsAlert":"BIG NEWS ALERT!","subscriptionText":"House of Math has introduced a subscription version","firstMonth":"First month","percentageOff":"90% off","fiveKr":"5 Kr","couponUpdated":"Get \u003csemiBold\u003e90% off your first month \u003c/semiBold\u003e as a huge thank you for your amazing support!\u003csuccessImage /\u003e \nSubscribe now!","notNow":"Not now","getMyOffer":"Get my offer","streak":"Streak","don'tShowAgain":"Don't show again","newDashboard":{"title":"Hey!","titleMain":" We updated your home area","description":"Don't worry, all your progress and belongings are still here","buttonText":"Okay"},"notifications":"Notifications","noMessage":"No Messages","justNow":"Just now","showMore":"Show more"},"errors":{"chooseAtLeastOneValue":"Choose at least one value","fieldRequired":"Field required","incorrect_password":"Wrong password","invalid_email":"Please give a valid email","noUser":"Don't have an account?","passwordIncorrect":"Wrong password","unknown":"Something went wrong. We are working on a solution. Try again later.","userDoesNotExist":"There are no users with this email address","emailInUse":"The provided email address is already in use. Try to log in.","emailInvalid":"The provided email address is invalid","authErrors":"authErrors group","email_validation_invalid":"Link invalid","email_validation_expired":"Link expired","write-above-this-line":"write above this line"},"campaign":{"pageTitles":{"landingPage":"Back to school offers"},"metaDescriptions":{"landingPage":"Back to school offers campaign"},"title":{"studyTips":"Free study tips","tutorOnDemand":"Chat with a Math Tutor 24/7 for free"},"tutorPackage":{"hours_4":"4 hours private tutoring","hours_8":"8 hours private tutoring","hours_16":"16 hours private tutoring"},"subtitle":"Sign up for free instant access","alreadyHaveAccount":"Already have an account?","accessWithEmail":"Continue with Email","accessWithGoogle":"Continue with Google","accessWithFacebook":"Continue with Facebook"},"common":{"userTypes":{"pupil":"Pupil","student":"Student","teacher":"Teacher","parent":"Parent","grandparent":"Grandparent","enthusiast":"Enthusiast","other":"Other","pupilDescription":"Pupil","studentDescription":"Student","teacherDescription":"Teacher","parentDescription":"Parent","grandparentDescription":"Grandparent","enthusiastDescription":"Enthusiast","otherDescription":"Other"},"typeform":{"takesOneMin":"Takes only 1 minute","heading":"We want your opinion!","description":"Help us become even more awesome for YOU!","buttonText":"Start"},"gradeLevels":{"1":"1st grade","2":"2nd grade","3":"3rd grade","4":"4th grade","5":"5th grade","6":"6th grade","7":"7th grade","8":"8th grade","9":"9th grade","10":"10th grade","11":"11th grade","12":"12th grade","13":"11th grade","14":"12th grade","15":"13th grade","16":"12th grade","17":"13th grade","noGrade":"No grade"},"submit":"Submit","last_item":"add above this"},"bootcamp":{"activityPage":"Activity","algebra":"Algebra","avatarAlts":{"vibeke":"Vibeke's avatar in a circle","jaggesher":"Jaggesher's avatar inside a circle","jarle":"Avatar for Jarle inside a pink circle","mari":"Mari's avatar in a colorful circle","monir":"Avatar for Monir within an orange circle","truc":"Avatar for Truc"},"backToBootcamp":"Back to bootcamp","backToOverview":"Back to overview","backToLearningPath":"Back to Learning Path","bootcamps":"Bootcamps","backToCurriculum":"Back to Curriculum","bootcampWarningMessageCorrFactor":"Expanded by the correct factor","bootcampWarningMessageCorrFraction":"Written the correct fraction","bootcampWarningMessageCorrMult":"Multiplied the numbers correctly","bootcampWarningMessageSameFraction":"Expanded by the same factor above and below","bootcampWarningMessageTitle":"Something went wrong here, make sure you have:","categorySubtitle":"These are the main topics you can work on","chatWithTeacher":"Chat with a Tutor","comingSoon":"Coming soon","completed":"Completed","continue":"Continue","congratulations":"Congratulations!","continueCurrent":"Continue with this activity set","stay":"Stay","correct":"Correct","earnedGems":"Earned gems","functions":"Functions","feedback":{"select-correct":"Only select the correct alternatives","select-all-correct":"Select all correct alternatives","select-only-correct":"Only select the correct alternatives"},"gems":"Gems","geometry":"Geometry","goNextActivity":"Go to next activity set","nextActivity":"Next activity","goodJob":"Good job!","youCanDoBetter":"You can do better","goToActivitySet":"Go to Activity Set","gotoBootcamp":"Go to bootcamp","gotoSubjectOverview":"Go to subject overview","gymsHint":"Please pick whatever topic you would like to work on. If you are unsure follow the number path","headingAlgebra":"Subjects in Algebra","headingGeometry":"Subjects in Geometry","headingNumbersAndQuantities":"Subjects in Numbers and Quantities","headingStatisticsAndProbability":"Subjects in Statistics and Probability","headingFunctions":"Subjects in Functions","headingProofs":"Subjects in Proofs","help":"Help","how":"How?","howPage":"How","howTitle":"How...","if":"If","ifYouKnowthat":"If you know that","inProgress":"In progress","interactiveTasks":"Interactive tasks","level":"Level","levels":"levels","limitedFreePromo":"Free for a limited time","metaDescriptionBootcampCategory":"Online math courses with animated instructional videos and interactive problem solving. Learn the K-12 curriculum in a fun, gamified and easy to understand way.","metaDescriptions":{"numbers-and-quantities":"Online math courses in arithmetic and number sense with animated instructional videos and interactive problem solving. Learn in a fun, easy and gamified way.","algebra":"Online math courses in algebra. Learn about variables, equations and inequalities through animated educational videos and interactive exercises in a fun way.","geometry":"","functions":"","proofs":"","statistics-and-probability":""},"module":"Module","needHelp":"Do you need more help?","next":"Next","numbersAndQuantities":"Numbers and Quantities","overview":"Overview","possibleActivityGems":"You have earned all possible gems for this activity set","possibleLevelGems":"You have earned all possible gems for this level","prerequisites":"Prerequisites","privateTutoring":"Private Tutoring","proofs":"Proofs","questions":"Tasks","quit":"Quit","readMore":"Read more","retake":"Retake","start":"Start","statisticsAndProbability":"Statistics and Probability","then":"Then","thenWhatIsMissingHere":"Then what is missing here?","title":"Bootcamps in Math | Interactive Exercises","tryAgain":"Try again","tutoring":"Tutoring","understood":"Understood","tip":"Tip:","videoUnderstoodTooltip":"If you know the content of the video, click here to mark it as completed.","understoodTooltip":"If you understand this, click here to mark it as completed.","usefulWords":"Useful Words","otherResources":"Other resources","videoAndActivities":"Videos \u0026 Activities","menu":"Menu","video":"Video","videos":"Videos","activity":"Activity","videoChatTutor":"Video Chat with Tutor","why":"Why?","whyPage":"Why","whyTitle":"Why...","wrong":"Wrong","signs":{"division":"÷","decimal":"."},"imageAlts":{"headerVineTablet":"Green beanstalk vine stretching from left to right","purpleArrowLeft":"A small purple bracket pointing to the left","greenLeaf":"Leaf","greenCheckMark":"Green checkmark","zoomInButton":"Zoom in button","zoomOutButton":"Zoom out button","closeButton":"Close button"},"upcomingTagAlt":"Purple sign hanging on a nail that reads Coming Soon","engVideoTag":"This video is currently only available in English","engVideoTagLongDivision":"This video will soon be updated with Norwegian method","badgeAltText":"badge","fractions":{"number_placed_in":"The number {{ value }} is placed in","numerator":"Numerator","denominator":"Denominator","illustrated":{"how_many":"Write the number of {{ item }} in the numerator, and the total number of {{ group }} in the denominator.","group":{"marbles":"marbles","balls":"balls","cubes":"cubes","animal":"animals","houses":"houses"},"item":{"redHouses":"red houses","blueHouses":"blue houses","redCubes":"red cubes","blueCubes":"blue cubes","redMarbles":"red marbles","blueMarbles":"blue marbles","basketballs":"basketballs","footballs":"footballs","dogs":"dogs","cats":"cats"}},"properFraction":"Proper fraction","improperFraction":"Improper fraction","mixedFraction":"Mixed fraction","largest":"largest","smallest":"smallest","comparing_text":"Click on the {{ largestSmallest }} fraction","undefined":"undefined","comparing_figure":"Click on the figure representing the {{ largestSmallest }} fraction","yes":"Yes","no":"No"},"geometryActivities":{"line":"Line","ray":"Ray","line_segment":"Line segment"},"subject":"subject","subjects":"subjects","headingCategories":"Bootcamps","helpContents":{"categories":"Watch animated videos and solve interactive exercises about central subjects in math.","algebra":"Watch animated videos and solve interactive exercises about central subjects in algebra.","geometry":"Watch animated videos and solve interactive exercises about central subjects in geometry.","numbersAndQuantities":"Watch animated videos and solve interactive exercises about central subjects in numbers and quantities.","statisticsAndProbability":"Watch animated videos and solve interactive exercises about central subjects in statistics and probability.","functions":"Watch animated videos and solve interactive exercises about central subjects in functions.","proofs":"Watch animated videos and solve interactive exercises about central subjects in proofs."},"valueOf":"The value of {{variable}} is {{value}}","previous":"Previous","activities":"Activities","activitySet":"Activity Set","activitySets":"Activity sets","new":"New","updated":"Updated","availableInEn":"Available in English","onlyInEn":"Only in English","noOfBootcamp":"video course","videoGenre":"Educational \u0026 Mathematics","youGot":"You got","onYourBestAttempt":"on your best attempt","nextLevel":"Next level","menuBossBattle":"Boss Battle","retakeActivitySet":"Retake the activity set","greatJobRetakeActivity":"Great job! Do you wish to retake or go to next?","mathTopics":"All Math","topicVideoActivityCount_zero":"{{noOfWatchedVideos}}/{{count}} Video \u0026 {{noOfActivities}} Activity sets","topicVideoActivityCount_one":"{{noOfWatchedVideos}}/{{count}} Video \u0026 {{noOfActivities}} Activity sets","topicVideoActivityCount_other":"{{noOfWatchedVideos}}/{{count}} Videos \u0026 {{noOfActivities}} Activity sets","done":"Done","levelCompleted":"Level Completed!","welldone":"Welldone! You have completed the last activity set","correctPercentage":"{{percent}}% Correct"},"subscription":{"page":{"title":"Subscribe","subscribeNow":"Subscribe now","unlockFull":"Unlock full access!","metaDescription":"Easy, fast and effective ways to improve your math grades. Learn math quick and easy. Subscribe now to access our bootcamps and grow your math skills."},"header":{"title":"Learn math smarter and easier","joinToGetAccess":"Join to get access","login":"Log in","buyNow":"Buy now","joinNow":"Join now","homTeam":"House of Math Team","freeTrial":"Free Trial","basic":"Basic","priceStartsFrom":"Price starts from","getBetterGrade":"Get a better math grade","totalMathSolution":"Say hello to your total math solution","trustedByTwoMillion":"Trusted by 2+ million \u003cbr\u003e registered users","faq":"Frequently Asked Questions","subscriptionSection":"Get a better math grade"},"pricing":{"productTitleYearly":"Year","perMonth":"per month","regularBilling":"Billed on a regular basis","title":"Get a better math grade","subTitle":"From ages 6 to 100+","month":"month","year":"year","free":"Free","per":"per","signUpForFree":"Sign up for free","continueForFree":"Continue for free","joinNow":"Join now","startNow":"Start now","changePlan":"Change plan","currentPlan":"Current plan","upgrade":"Upgrade","bestValue":"Best value","getAccessTo":"Get access to","signUp":"Sign up","cancelAnytime":"Cancel any time","couponPeriodPlural":"for {{ durationInMonths }} months {{ priceLabel }}","couponPeriodSingular":"for 1 month {{ priceLabel}}","couponOnce":"{{ priceLabel }} first {{ interval }}","afterTime":"Then","freeTrialPeriodPluralCheckout":"{{ freeTrialPeriod }}-day Free Trial","freeTrialPeriodSingular":"1 day free","freeTrialPeriodPlural":"{{ freeTrialPeriod }} days free","seeMore":"See More","close":"Close","showAllPackages":"Show all packages","showLessPackages":"Show less packages","textYearly":"{{ price }} charged yearly","testFreeTrialPeriod":"Start a {{ trialDays }}-day FREE Trial","startFreeTrialPeriod":"Start {{ trialDays }} day FREE Trial","unlockTrialPeriod":"Unlock your {{ trialDays }}-day Free Trial","then":"then","buy":"Buy","whatsIncluded":"See what is included","showLess":"Show less","familyAccountComingSoon":"Family Account coming soon"},"product":{"title":"Get your own personalized Learning Path","faq":"FAQ","faqLogo":"FAQ logo","teacher":"I am a teacher","products":{"videoCourses":{"title":"Your path to mastering math","tagline":"Bootcamps","text":"Digital courses with instructional videos and interactive problem solving"},"keepTrack":{"title":"Keep track","tagline":"Statistics panel","text":"Your own statistics panel to keep track of your progress"},"unlimitedQuestions":{"title":"Unlimited questions ","tagline":"24/7 math chat","text":"Ask our tutors unlimited questions in our math chat"},"excercises":{"title":"Everything you need in one place","tagline":"Exercises","text":"Gain access to the full K-12 curriculum"}}},"checkout":{"title":"Subscription checkout","metaDescription":"Checkout"},"success":{"confirmation":"Confirmation","title":"Success","metaDescription":"Success","date":"Order date","confirmationNumber":"Confirmation number","welcome":"Welcome to","fullAccess":"You now have access to House of Math!","goHomeButton":"Go to Home","backToUrl":"Back to where you left off","yourSubscriptionPackage":"Your subscription package","trialPeriodEndsOn":"Trial period ends on","packageUpgraded":"Package upgraded","packageChanged":"Package changed","checkmark":"Checkmark logo","getStarted":"Get Started"},"callback":{"text":"You're almost there - we are processing your subscription plan"},"paymentFailed":{"message":"We aren't able to process your payment.\nPlease try again.","tryAgain":"Try again","refused":"Payment declined","refusedSubtitle":"Please try again with another payment method or card.","cancelled":"Payment cancelled","error":"Payment failed","errorSubtitle":"An error occured while processing your payment.\nPlease try again later.","metaDescription":"Payment failed"},"paymentPending":{"title":"Payment processing","metaDescription":"Payment processing","message":"Your payment is being processed","messageEmailSent":"A confirmation e-mail will be sent to \u003cemail\u003e{{email}}\u003c/email\u003e"},"modal":{"title":"Start Free Trial","text":"Thank you for coming back to us.","encyclopediaText":"Register to get access to our complete math encyclopedia","trialUsed":"Subscribe now!","button":"Get unlimited access","freeVersion":"Continue to use free version","bulletPointsTitle":"Also get access to:","teacher":{"title":"Special teacher discount","trialEnd":"Your trial period has now ended, but we’re excited to offer you an exclusive opportunity to continue using House of Math","token":"As a token of our appreciation, we’d like to offer you \u003cstrong\u003e50% off\u003c/strong\u003e your subscription for \u003cstrong\u003e6 months\u003c/strong\u003e!","copyCode":"Copy the code above and add it in checkout for monthly package. Valid until february 15th 2025.","copy":"Copy","claimOffer":"Claim your discounted offer"},"dashboard":{"register":"Register","registerFree":"Register for free","fullAccess":"Want to get full access?","startFree":"Start a free trial"}},"onboarding":{"learningPath":"Creating your Learning Path","choosePlan":"Choose the plan that fits your needs"},"testimonial":{"explore":"Explore the how and why \u003cbr\u003e behind each concept","stepByStep":"Detailed step-by-step \u003cbr\u003e explanations","provenMethod":"Proven method for math \u003cbr\u003e success","reviewSilje":"I am very pleased with the follow-up I received. This lead to new accomplishment and I have gotten better grades","reviewThorbjorn":"I sent my daughter to House of Math when she started middle school, not because she struggled with math, but because she needed motivation. Her teenage brain needed guidance as well as accelerated math education. We are extremely satisfied with both!"}},"chat":{"expressionError":"Something went wrong with this expression","header":"Chat with Tutor","infoBox":"Hello! \u003cbr\u003e\u003cbr\u003e Please pose your math questions here. We may not be able to answer questions about university level math.","infoBoxEnd":"This chat service is free for a limited time only. The chat is only available in English.","infoBoxItem1":"1. Formulate a specific problem.","infoBoxItem2":"2. Provide all details.","infoBoxItem3":"3. Describe what you’ve tried.","infoBoxList":"Tips for getting help as quickly as possible:","infoMessage":"Pose a question to a tutor, we'll reply as quickly as we can.","tutorChat":{"alreadyClaimed":"Someone claimed this chat before you.","chatHeader":"User ID {{userId}}","claimThread":"Claim Chat","claimWarning":"By Claiming this chat, you are the one that will answer it. It is your responsibility to give a complete answer.","end":"End Chat","endWarning":"When you end the chat, you can`t answer this person anymore.","listActiveChat":"My Active Chats","listHeader":"Messages","listQueue":"Queue","noActiveChat":"You have no active chats","noNewMessages":"There are no new messages","offline":"Offline","online":"Online","onlineStatus":"Online Status"},"newMessage":"Pose your question here","noTeachersAvailable":"There are no tutors available at the moment","onlineTeachers":"Tutors online","openingHours":"Open 24/7 \u003cbr\u003e Chat in English","receivedMessage":"We have received your message and will reply as soon as possible!"},"chatAi":{"consentText":"Hey! I'm Oracle, your math AI. I'm here to help you learn math.","ctaAsk":"Ask \u003cbr /\u003ethe Oracle","ctaLearningPath":"Go to Learning Path","askHelp":"How can I help you?","networkError":"Sorry, something went wrong. Please try agin later.","unauthorizedError":"Unauthorized","rateError":"Uh-oh! Looks like you've used up your daily quota. Upgrade to continue.","rateErrorSubscribed":"Looks like the Oracle is in high demand right now. Check back later.","exitOracle":"Exit","placeholder":"Ask me about math...","subscribe":"Subscribe","betaString":"Beta","globeAlt":"Globe AI","oracleImageAlt":"Oracle Image","unauthorizedMessage":"Start a free trial and I will answer your question","user":"User","backToActivity":"Back to Activity"},"userFeedback":{"imageAlt":{"veryDissatisfied":"Very Dissatisfied","dissatisfied":"Dissatisfied","neutral":"Neutral","satisfied":"Satisfied","verySatisfied":"Very Satisfied","placeHolder":"Select a rating","close":"close","feedbackButton":"Give feedback"},"header":"Got any feedback?","commentBox":"Write your comment here (optional)","feedbackButtonText":"Feedback","sendButtonText":"Send","errorMessage":"Something went wrong"},"gemShop":{"buyNow":"Buy now","comingSoon":"Coming soon","itemTypes":{"chest":"Chest","head":"Head","neck":"Neck","hats":"My headwear","jewelry":"My jewelry","clothes":"My clothes","special":"Special"},"menu":{"pet":"Pet items","featured":"Featured"},"noGems":"Need more gems","owned":"Owned","store":"Store","tier":"Tier","upgrade":"Upgrade","upgradeTypes":{"productionRelativeIncrease":"Gem boost +{{value}}%","productionAbsoluteIncrease":"Gem boost +{{value}}","tickRateIncrease":"Timer speed {{value}}x","energyCapacityIncrease":"Kudos capacity +{{value}}"},"unlockRank":"Dragon Rank {{rank}}","rankUp":"Rank up your dragon to unlock this item","padlockAlt":"Padlock","allItems":"All Items"},"homebase":{"mainView":"Main","dragonView":"Dragon","storeView":"Store","cavern":"Dark Cavern","checkBaseForQuests":"Check out your Learning Path","chestText":"This is your base. Return here to view math tasks and updates for your mathematics journey. Out there you'll find things such as this:","chestTitle":"Welcome!","close":"Close","collection":"My Collection","collectionList":{"allItems":"All Trophies","allRewards":"All rewards","bootcamps":"Bootcamps","competition":"Competition","mathdrill":"Math Essentials","mathEssentials":"Math Essentials"},"collectionShowcase":{"bootcamp":"Bootcamps","gathered":"Gathered","championship-2021":"Multiplication Championship 2021"},"weExchange":"We exchange","completed":"Completed","rankPopupTitle":"New XP level-up system!","rankPopupText":"You can earn XP by doing activities in the portal. Reaching the XP threshold will make you level up.","dummyData":{"mult":{"title":"Multiplication","description":"What is Multiplication?"},"algebra":{"title":"Algebraic expressions","description":"Elementary Algebra"}},"event":{"checkQuest":"I have read and understood","INTRO":{"dialogue":[{"text":"This is your base. Here you will receive messages and challenges while you collect treasures and rewards during your journey through the world of mathematics.","title":"Welcome home"}]}},"findYourLearningPath":"Find your Learning Path, create an avatar and get a pet dragon!","gemBlueAlt":"Sparkling blue-white diamond","gemGreenAlt":"Sparkling green emerald","gemRedAlt":"Shining red ruby","gotoavatarpage":"Go to avatar","Home":"Homebase","homebasepage":"Start Base:","imageAlt":{"goldenHex":"Golden hexagon","levelBadge":"Level badge"},"introText":"You're exactly the hero we're looking for. We'll return to that. Let me show you around first.","introTitle":"Good start!","kudos":"Kudos","level":"Level","levelUp":"Level up","navbarIcon":"Space","next":"Next","shadow":"Shadow","playerName":"Player name...","questCompleted":"Math Task Completed!","questDescriptions":{"bootcamp":"Explore the \u003cb\u003e{{BootcampTitle}}\u003c/b\u003e bootcamp and defeat the boss!","essential":"Complete the Math Essential to prove your skills!","encyclopedia":"Read and understand \u003cb\u003e{{EncyclopediaArticle}}\u003c/b\u003e and tick the checkbox!"},"quests":"My Math Task","mathQuests":"Math Tasks","completedMathQuests":"Completed Tasks","completeQuestsToSee":"Complete your tasks to see them here","questText":"Take the mapping quiz to get your custom math journey","quiz":{"advancedQuizText":"To get more math tasks, pick the subjects that you would like to learn more about!","advancedQuizTitle":"Good work!","applyText":"You obviously know a lot about math! If you want to join the House of Math team, please apply \u003clinkToCareers\u003ehere\u003c/linkToCareers\u003e.","awardTitle":"We gamify your experience","awardTitleSubtext":"On the journey, you will find reward chests like this","awesome":"Awesome!","buttonTakeQuiz":"Take quiz","creatingYourLearningPath":"Creating your Learning Path...","learningPath":"Creating your Learning Path","subjectKnowledgeRating":{"0":"Poorly","1":"A bit poorly","2":"Neutral","3":"Well","4":"Very well"},"curriculumSubjectKnowledgeRating":{"1":"Not at all","2":"Slightly","3":"Moderately","4":"Quite Well","5":"Very well"},"title":"For effective learning, take the quiz!","titleSubtext":"Take this quiz to find out where to start.","yesNoQuestionTitle":"Have you ever worked on \u003clowercase\u003e{{ subject }}\u003c/lowercase\u003e?"},"retakeQuiz":"Retake Quiz","reward":"Reward","storyCards":{"buttonCool":"Cool!","buttonIWill":"I will!","buttonJump":"Jump","buttonLetUsTry":"Let us try!","buttonNice":"Nice!","buttonOfCourse":"Of course!","buttonOh":"Oh!","buttonPeekaBoo":"Peeka Boo","buttonRoar":"Roar","buttonThanks":"Thanks!","buttonWow":"Wow!","buttonYes":"Yes!","ChoosePetTrick":"Your dragon is ready to learn a new trick! Pick one of the three below!","ChooseYourEgg":"Look, someone has left a gift for you!","CoolDown":"You need to wait a while before you can give Math Energy again! Use your time to collect more Energy Orbs from the Math Essentials!","CoolDownFirstTime":"Your Egg is full, and you need to wait a while before you can give more Math Energy! Use your time to collect more Math Energy from the chests in Math Essentials!","DoNotHaveEnoughMnO":"You do not have any more Kudos! Do math tasks to gather more!","DragonFull":"Your dragon is full! Wait a while before giving more Kudos.","DragonTrickCooldown":"You dragon is tired after training for the new trick. Give it some rest before trying again!","DragonTrickNotWilling":"You dragon doesn't want to train until it's happy and fed!","DragonTutorial1":"Keep your dragon healthy by feeding it Kudos!","DragonTutorial2":"Rumors say that if you keep your dragon healthy for a while, they might learn new tricks!","DragonTutorial3":"Dragons also need affection! Pet your dragon regularly!","DragonTutorial4":"Nice! Your dragon is happy! Continue to give it affection every day!","EggHasBeenChosen":"Eggcellent choice! Let's place the egg safely in your base. Can we make it hatch?","EggHungry1":"Hmm... Your egg seems hungry!","EggHungry2":"Here take this Kudos!","EggHungry3":"Try feeding Kudos to the egg by clicking on it!","EggInfoStillHatching":"Your egg is currently hatching! However, you need to feed it more Kudos!","EggIsAboutToHatch":"Incredible! You now have got a dragon in your base!","FeedMathEnergyToEggFirstTime":"Try feeding Kudos to the egg by clicking on it!","FirstTimeGettingMathEnergyFromChest":"Cool! You discovered your first Kudos! Give it to your egg in the Base!","GiveDragonName":"Type in your dragon's name or simply press OK.","InfoAboutPetTrick":"Your dragon is willing to learn when it's fully fed and happy!","MathEnergyTutorial1":"You need to feed enough Kudos to the egg in order to make it hatch.","MathEnergyTutorial2":"Do math tasks in order to find more Kudos!","NextHatchingState1":"Wow! Look! Your egg looks different now!","NextHatchingState2":"Your egg has hatched some more! Wonder what's inside...","NextHatchingState3":"Just a little bit more... Your egg is hatching soon!","TimesTableCompetition2021FirstPlace":"Congratulations on winning the 2021 Times Tables Competition! Claim your unique trophy to add it to your collection.","TimesTableCompetition2021Participant":"Amazing job! Your earned this trophy by participating in the 2021 Times Tables Competition.","TimesTableCompetition2021SecondPlace":"Congratulations on 2nd place in the 2021 Times Tables Competition! Claim your unique trophy to add it to your collection.","TimesTableCompetition2021ThirdPlace":"Congratulations on 3rd place in the 2021 Times Tables Competition! Claim your unique trophy to add it to your collection."},"space":"Space","takeTheQuiz":"Take the quiz to unlock your Home!","tooltip":{"videoCourse":"Bootcamp","encylopedia":"Encyclopedia","mathEssential":"Math Essential","description":{"videoCourse":"Watch videos, do activities and defeat the boss","encylopedia":"Read and understand, then tick the checkbox.","mathEssential":"Complete the Math Essential to prove your skill!"}},"titles":{"houseOfMathMember":"House of Math Member","mathApprentice":"Math Apprentice","mathemagician":"Mathemagician","mathJunior":"Math Junior","mathlete":"Mathlete","mathScholar":"Math Scholar","mathSorcerer":"Math Sorcerer"},"unlockYourHomebase":"Log in or sign up to unlock your dashboard, where you can create your personalized Learning Path, see your XP and rewards, access your Space and much more!","updateQuests":"Update math tasks","xp":"XP","open":"Open","removeTask":"Remove From Learning Path","cancel":"Cancel","rank":"Rank","leagueNavButton":"League","menuLabel":"Menu","loading":"Loading","loadscreenHint":{"feedDragon":"Rank up your dragon by feeding it kudos","earnKudos":"Earn Kudos by doing math activities","earnGems":"Earn gems by doing math activities","buyStuff":"Buy cool stuff with gems in the dragon shop","wardrobe":"You can find all your bought items in the wardrobe","moreMath":"The more math you do, the more gems and Kudos you'll get"}},"menu":{"mathTopics":"All Math","bootcamp":"Bootcamps","dashboard":"Home","digitalTools":"Digital Tools","drillhefter":"Math Essentials","encyclopedia":"Encyclopedia","exams":"Eksamensoppgaver","exitPreview":"Exit preview","games":"Games","logOut":"Log Out","login":"Log in","mathEssentials":"Math Essentials","mathVault":"Junior Math","register":"Sign up","settings":"Settings","studyTips":"Study Tips","toggleNavigationMenu":"Toggle navigation menu","tutors":"Tutor-on-Demand","daysStreak":"{{days}} day streak","learningTools":"Learning Tools","moreLearningTools":"More Learning Tools","penNPaperExercise":"Pen \u0026 Paper exercises","excelOrGeoGebra":"Excel / GeoGebra recipes","numberNQuantities":"Numbers \u0026 Quantities","statNProbability":"Statistics \u0026 Probability","stackNLoad":"Stack n´load","juniorMath":"Junior Math","windSurfer":"Wind Surfer","treasureTrail":"Treasure Trail","multiplicationMaster":"Multiplication Master","misc":"Profile","showMore":"Show more","showAllGames":"Show all games","gotoGames":"Go to games","getHomApp":"Get the House of Math app","tryFree":"Start free trial"},"navbar":{"explore":"Explore","menu":"Menu","panelAdmin":"Admin Panel","panelTutor":"Tutor Panel","panelStudent":"Student Panel","search":"Search","languages":"Language","norwegian":"Norsk","english":"English","ukrainian":"Українська","settings":"Settings","logOut":"Log Out","dashboard":"Home","chat":"Chat","userProfile":"Profile","leaderboard":"Leagues"},"notFound":{"clickOne":"Click the one that needs to be fired","fire":"Fire","goDashboard":"Go to Dashboard","pageNotFound":"Page not found","pageRemoved":"Page removed","punishDevs":"Our development team must be punished for this huge failure","employeesOnLunchBreak":"It seems our employees have taken a lunch break. We’ll be back soon!"},"search":{"adaptedView":"Customized View","bootcamp":"Bootcamps","all":"All","drill":"Math Essentials","encyclopedia":"Encyclopedia","geogebra":"GeoGebra","noResult":"No search result found","readMore":"Read more","watchVideo":"Watch video","remove":"Remove","search":"Search..","searchResultAppear":"Your search results will appear here","searchTooltip":"Customized View shows you entries for the level you chose in settings. Toggle to see all entries.","software":"Digital tools","title":"Search results","searchPlaceholderText":"What would you like to learn?"},"dashboard":{"activities":"Activities","activity":"Activity set","activitiesShownHere":"Your latest visited pages will be shown here","activitiesSubtitles":{"multgame":"Last mode","stacknload":"Last level","treasureTrail":"Last level","windSurfer":"Last score"},"bootcamp":{"buttonContinueLearning":"Continue learning","buttonToOverview":"Go to overview","comingSoon":"Coming soon!","edit":"Edit","title":"My Bootcamps"},"chatBetweenUserAndTutor":"Chat between user and tutor","checklists":{"button":{"claimReward":"Claim reward","completeProfile":"Complete profile","createAvatar":"Create Avatar","exploreLeaderboard":"Leaderboard","exploreSpace":"Take me to Space","mathTask":"My Math Tasks","onboarding":"Mapping Quiz","verifyEmail":"Verify E-mail"},"checklist_item_completed":"Checklist Item Completed","close":"Close","completeProfile":"Complete profile","continue":"Continue","topicOverview":"Topic overview","createAvatar":"Create avatar","description":"Complete your checklist and be rewarded","exploreLeaderboard":"Visit the league","exploreSpace":"Visit your space","imageAlt":{"completeProfile":"Profile icon","createAvatar":"Avatar icon","exploreLeaderboard":"Trophy icon","exploreSpace":"Space icon","mathTask":"Math tasks icon","notification_checkbox":"Checklist notification checkbox","onboarding":"Brain icon","verifyEmail":"E-mail icon"},"mathTask":"Complete a Math Task","mathTaskBadge":"Beat the boss","notification":{"completeProfile":"You have completed your profile.","createAvatar":"You have created your Avatar.","exploreLeaderboard":"You have visited the league.","exploreSpace":"You have entered your Space.","mathTask":"You have completed your first math task.","onboarding":"You have completed your mapping quiz.","verifyEmail":"You have verified your e-mail address.","pickCurriculum":"You have picked your curriculum.","watchVideo":"You have watched a video.","completeCurriculumActivity":"You have completed a curriculum activity set."},"seeMore":"See more","title":"Get started","completeChecklist":"Complete the Checklist","pickCurriculum":"Pick your curriculum","watchVideo":"Watch a video","completeCurriculumActivity":"Complete a curriculum activity set","tooltip":{"completeProfile":"\u003cstrong\u003eComplete\u003c/strong\u003e your profile and be rewarded \u003cstrong\u003e{{rewardXP}}{{comma}}{{rewardGems}}.\u003c/strong\u003e","createAvatar":"\u003cstrong\u003eCreate\u003c/strong\u003e your personalised avatar and \u003cstrong\u003eclaim {{rewardXP}}{{comma}}{{rewardGems}}.\u003c/strong\u003e","exploreLeaderboard":"\u003cstrong\u003eVisit\u003c/strong\u003e the league and \u003cstrong\u003eearn {{rewardXP}}{{comma}}{{rewardGems}}!\u003c/strong\u003e","exploreSpace":"\u003cstrong\u003eClaim {{rewardXP}}{{comma}}{{rewardGems}}\u003c/strong\u003e by exploring your \u003cstrong\u003eSpace\u003c/strong\u003e","mathTask":"\u003cstrong\u003eComplete\u003c/strong\u003e your first math task by beating \u003cstrong\u003ethe boss.\u003c/strong\u003e","onboarding":"\u003cstrong\u003eEarn {{rewardXP}}{{comma}}{{rewardGems}}\u003c/strong\u003e by completing the \u003cstrong\u003emapping quiz.\u003c/strong\u003e","verifyEmail":"\u003cstrong\u003eVerify\u003c/strong\u003e your e-mail and claim \u003cstrong\u003e{{rewardXP}}{{comma}}{{rewardGems}}.\u003c/strong\u003e","rewadXP":"{{XP}} XP","rewadGems":"{{GEMS}} gems"},"verifyEmail":"Verify e-mail"},"checkOutMath":"Check out some of our math topics","completed":"Completed","completeProfile":{"button":"Complete your registration","description":"We need to know a little bit more about you to customize your experience"},"doMath":"Do Math","doMathLT":"Do math!","featureProducts":{"label":"Main Math Topics"},"gamification":{"badge":"Bootcamp badge","badgeBackground":"Badge background","buttonToBase":"Space","completeQuestsGetTopics":"Complete tasks to get more topics","gemIcon":"Gem icon"},"imageAlt":{"background":"Green beanstalk with many leaves, twisting dramatically","backgroundMobileBottom":"Bean stalk","backgroundMobileTop":"Bean stalk","homebasePlaceholder":"House of Math logo with animated figures on top"},"latestActivities":"Last visited","learningPath":"Learning Path","learningPathTutorial":{"step1":{"title":"Based on your answers, \u003cbr /\u003e this is your \u003cspan\u003eLearning Path\u003c/span\u003e","action":"Next"},"step2":{"title":"To complete a level, watch the \u003cbr /\u003e \u003cspan\u003evideos\u003c/span\u003e and do the \u003cspan\u003eactivities\u003c/span\u003e","action":"Tell me more"},"step3":{"title":"Check your learning by \u003cbr /\u003e completing the \u003cspan\u003eBoss Battle\u003c/span\u003e","action":"Next"},"step4":{"title":"You can always get new tasks by updating your \u003cspan\u003eLearning Path\u003c/span\u003e","action":"Next"},"step5":{"title":"To get an overview of all our content, go to \u003cspan\u003eAll Math\u003c/span\u003e","action":"Finish"}},"curriculumTutorial1":{"step1":{"title":"Each cap \u003cCapImage /\u003e represent a math topic. Collect them all to master the topic.","action":"Next"},"step2":{"title":"Once all caps \u003cCapImage /\u003e are collected, you have mastered {{subjectLabel}} and can claim your mastery certificate","action":"Next"},"step3":{"title":"You can always create your personal Mastery Track here for the most efficient  math learning method","action":"Got it"}},"curriculumTutorial2":{"step1":{"title":"Each piece \u003cPuzzleImage /\u003e represents a math topic. Collect them all to open the Bonus Chest \u003cChest /\u003e","action":"Next"},"step2":{"title":"After collecting the puzzle pieces \u003cPuzzleImage /\u003e you can start mastering {{subjectLabel}} and collect the caps. \u003cCapImage /\u003e","action":"Next"},"step3":{"title":"Once all caps \u003cCapImage /\u003e are collected. you have mastered {{subjectLabel}} and can claim your Mastery Certificate","action":"Got it"}},"avatarTutorial":{"step1":{"title":"Click here to \u003cspan\u003ecreate your avatar\u003c/span\u003e","action":"Finish"}},"learningPathCardTutorial":{"step1":{"title":"Click button to get \u003cbr /\u003e your own \u003cspan\u003eLearning Path\u003c/span\u003e","action":"Ok"}},"mathTask":{"step1":{"title":"\u003cspan\u003eDefeat a Bootcamp Boss\u003c/span\u003e in order to complete a \u003cspan\u003eMath Task!\u003c/span\u003e","action":"Ok"}},"step":"Step","metaDescription":"All the math help you need for free! Get the full mathematics curriculum with instructional videos, interactive problem solving, games and 24/7 chat.","myActivities":"My Activities","news":{"buttonChat":"Chat","buttonPlay":"Play","buttonShowMe":"Show me","buttonStart":"Start","campAlt":"Campsite","content1":"Join D’Rillo on an adventure through the updated Math Essentials. Learn about the most central topics in mathematics while you play with fun activities!","content2":"Our tutors are ready to help you. Use the chat to get answers to all your questions about math.","content3":"Enter the Times Tables competition, compete against others and win great prizes!","content4":"Bootcamps are fun, interactive, gamified and animated math lessons for everyone.","content5":"Get instant math help! Book a 15 minute video session with a tutor.","header1":"Become a Drillmaster","header2":"Get Unstuck","header3":"Win an iPad and other great prizes","header4":"Try the Bootcamps","header5":"Video Chat","subject1":"Fractions","title":"News"},"newsInHomebase":"News in Homebase","reward":"Reward","statistics":{"activity":{"correct":"Correct","inCorrect":"Incorrect","instruction":"Try our products to unlock this statistic","label":"Total activities"},"feedback":{"instruction":"Would you like to see more statistics in the future?","no":"No","yes":"Yes"},"label":"My Statistics","timeSpent":{"days":"Days","hours":"Hours","label":"Time spent","minutes":"Minutes"}},"tagNew":"New","takeQuiz":"Take the quiz","tipOftheDay":"TIP OF THE DAY","title":"Dashboard","toolTip":"Overview of your progress and Learning Path.","tutor":{"assigningTutor":"Please wait while we are finding a tutor for you.","bookPrivateTutoring":"Book Tutoring","myStudentPanel":"My Student Panel","myTutor":"My tutor","myTutors":"My tutors","tutorPending":"Pending","tutorScheduling":"When you have scheduled a session, it will show up here."},"exploreMathTopics":"Explore Math Topic","userCard":{"quizMessageButton":"Quiz","quizMessageText":"High math competence expands your options in your future. The more math you know, the more options you'll receive. So, muster up your courage and get started on the math! Get started by taking the quiz!","quizMessageTitle":"Mapping quiz","youHaveNews":"You have news! \u003copenHomebase\u003eOpen homebase\u003c/openHomebase\u003e"},"yourPath":"Your Learning Path","completeLevel":"Complete \u003cb\u003elevel {{level}}\u003c/b\u003e to unlock","gotIt":"Got it","lockedChest":"Chest is locked","congratulations":"Congratulations!","collectRewards":"You got more \u003cb\u003eKudos\u003c/b\u003e and \u003cb\u003egems\u003c/b\u003e to spend in your personal \u003cb\u003eSpace.\u003c/b\u003e","collect":"Collect","bossBattle":"Boss battle","challenge":"Challenge Boss","goToVideo":"Go to video","goToActivity":"Go to activity","goToBoss":"Go to Boss","startHere":"Start here","read":"Read","goToArticle":"Go to article","skip":"Skip ahead","completeTask":"Complete Math Task","task":"Task","readyProgress":"Ready to progress?","defeatBoss":"\u003cb\u003eDefeat\u003c/b\u003e the boss to \u003cb\u003ecomplete\u003c/b\u003e this Math Task!","completeBossBattle":"Complete task","updateLearningPath":"Update Learning Path","learningTracks":"Learning Tracks","intelligentPath":"Intelligent Learning Path","mathGaps":"Fill math gaps","getAPath":"Get an Intelligent Learning Path","latestLearning":"Latest Learning","addTrack":"Add track","removeTask":"Remove from learning tracks","noLearningYet":"Your latest math discoveries awaits! Go do math!","addCurriculum":"Add your curriculum","ultimateLearningTrack":"Ultimate Learning Track","ltChangeButton":"Change","stats":"Stats","viewStats":"View my stats","myStats":"My Stats","weeklyStats":"Weekly Stats","changeTopic":"Change topic","viewTopics":"View topics","learnFasterEasier":"Learn Math \u003cbr/\u003eFaster and Easier","latestMasteryPath":"Latest Mastery Path"},"gamification":{"questSummary":{"pointsEarned":"Points earned","reward":"Reward","starsEarned":"Stars earned","title":"Summary"},"globalLeaderboard":"Global Leaderboard","rank":"Rank","title":"Leaderboard","weekNo":"Week {{ weekNo }}","youEarned":"You earned","youJoined":"You just entered this week's \u003cstrong\u003e{{league}}\u003c/strong\u003e","league":{"bronze":"Bronze League","silver":"Silver League","gold":"Gold League","platinum":"Platinum League","diamond":"Diamond League"},"congratulations":"Congratulations","bravo":"Bravo!","ohNo":"Oh no!","youDemoted":"You demoted to","youStayed":"You held your position in the","yourRank":"with rank \u003cstrong\u003e#{{rank}}!\u003c/strong\u003e","advancing":"You advanced to","gems":"Gems","badgeUnlock":"Badge Unlocked!","openCollection":"Click to open Badge Collection","bossDefeated":"Bootcamp Boss defeated!","copied":"Code copied","specialDiscount":"SPECIAL DISCOUNT!","shareCode":"Share the code for 30% off","copy":"Copy","accountUpgrade":"Your Account has been upgraded!","freeAccess":"As a teacher you have free access to all our premium features."},"home":{"bookTutoring":"Book tutoring","heroDescription":"Math designed in a fun way to keep you engaged and focused.","letsGo":"Let's go","startNow":"Start now","jumpIntoLearningToday":"Jump Into Learning Today","chooseLearningMethod":{"title":"We Cater to All Learning Styles","videos":"3D  animated videos","games":"Games","activities":"Interactive activities"},"highlightTexts":{"scientificallyProven":"Get a better grade","betterGrades":"with our scientifically proven method","reachDream":"Reach your dream faster","learningMath":"by learning math smarter and easier than ever before","ownPace":"Learn at your own pace","learningPath":"with our Personalized Learning Intelligence Frigg"},"testimonials":{"title":"We Love House of Math","siennaNorwayTestimonial":"House of Math stands out because you can go back to the basics without feeling ashamed. The platform builds understanding brick by brick and has given me the confidence to master math. Now I love learning and have seen a huge improvement in my grades.","mortenNorwayTestimonial":"House of Math makes math interesting for my three children and motivates them to learn. The teachers are great at explaining concepts, how things connect, and how math can be applied. The result is more independent children who perform better on tests, and a family with much less stress around homework.","catharinaNorwayTestimonial":"House of Math works like a private tutor that we always have access to. The AI bot explains the math problems, so we avoid the frustration when helping with homework. Math sessions have now become a positive experience for the whole family.","lisaNorwayTestimonial":"House of Math has videos and exercises that actually help me understand math. I had given up on the subject, but now I have better grades and the confidence to work on other subjects I struggle with as well.","andersNorwayTestimonial":"House of Math provides tailored learning that meets the child where they are. Homework time has become much less stressful, and our child now has the tools to find the answers on their own. We’re seeing a positive attitude toward math that has made everyday life easier for the whole family.","lisaAge":"15 years","namibia":"Namibia","srilanka":"Sri Lanka","india":"India","norway":"Norway","southAfrica":"South Africa","ghana":"Ghana","philippines":"Philippines","student":"Student","teacher":"Teacher","parent":"Parent","parentToThree":"Parent to three children"},"tutorOnDemand":{"title":"Do you want more than a digital experience?","subtitle":"Try Tutor-on-Demand","description":"Are you stuck on a math problem or have a question? \u003cbr /\u003e Chat live with a real math tutor and get the answers you need 24/7. \u003cbr /\u003e Or work with one of our 170 high-quality tutors. We are here to help you achieve your math goals and take the next step!"},"homPresentationTitle":"Our Cutting-Edge Approach","previewSite":"Sneak preview the site","products":{"mathTopics":"Find a complete overview of all math topics.","mathTopicsTitle":"All Math","bootcamp":"Learn from our online math courses with animated instructional videos and interactive problem solving.","bootcampTitle":"Bootcamps","digitalTools":"Learn to use Excel and GeoGebra for all sorts of calculations and visualizations.\n\n","digitalToolsTitle":"Digital Tools","encyclopedia":"Read everything in the mathematics curriculum from elementary school to high school.","encyclopediaTitle":"Encyclopedia","exams":"","examsTitle":"","games":"Get better at math while playing games and having fun!\n\n","gamesTitle":"Games","mathEssentials":"Get on the fast track through the most central topics in math!\n\n","mathEssentialsTitle":"Math Essentials","mathVault":"Practice more than 2,500 multiple choice questions from the K-4 curriculum.","mathVaultTitle":"Junior Math","tutorOnDemand":"Get all the help you need to learn math through live chat, video chat and tutoring with a tutor.","tutorOnDemandTitle":"Tutor-on-Demand","studyTips":"Discover tips on how to become a super student!\n\n","studyTipsTitle":"Study Tips","mappingQuizTitle":"Mapping Quiz","mappingQuiz":"Receive a personalized Learning Path based on our Mapping Quiz.","title":"Choose your favorite learning method","curriculumTitle":"Curriculum"},"sponsors":{"title":"Sponsoring our ESG project \u003cbr\u003e\u003c/br\u003e \u003cstrong\u003eWeTakeAction\u003c/strong\u003e"},"title":"Learn math smarter, faster and easier","pilotTitle":"Learn math smarter and easier!","pilotSubtitle":"We help you crack the math code\nLet's go to the next level","weAreGlobal":"Become a member of our \u003cstrong\u003eglobal\u003c/strong\u003e math network","whatAreYouLookingFor":"What are you looking for?","whoWillBenefit":{"title":"Who Benefits From Our Solution?","students":"Students","parents":"Parents","teachers":"Teachers","studentDescription":"Learn and practice math at your own pace with fun activities and games.","parentDescription":"Support your child’s math learning with tools you don't find in schools.","teacherDescription":"Enhance your teaching methods and learning outcomes."},"houseOfMathMethod":{"title":"House of Math’s Unique Teaching Method","method1":"Bite sized math learning","method2":"Access to all pre-university math","method3":"Fill gaps in your knowledge and thrive in class","method4":"Rooted in the teaching philosophy of the Nordics","method5":"Based on neuroscience, math didactics, and motivational psychology","method6":"Personalized and unique Learning Path designed by our AI","method7":"20 years empowering learners to become math champions"},"imageAlt":{"beanStalkWorld":{"mobile":"Leafy beanstalk wrapped around the whole world","desktop":"Beanstalk connecting planet Earth with an equals sign"},"hero":"House of Math Team","productBeanStalk":"Beanstalk linking a multiplication sign and a plus sign vertically","threeDragons":"Three dragons","mappingQuiz":"Brain with nodes","teamLogo":"House of Math Team logo"},"seeHow":"See how","signUp":"Sign up","used":"Used by","unlimited":"Unlimited","practiceExercise":"practice exercises","totalTeacher":"45 000+","teachersWorldwidePilotPage":"teachers worldwide","teachersWorldwide":"45 000+ teachers worldwide","makingSmarterKids":"20 years of helping kids","topMathLearners":"become top\u003cbr\u003e math learners","moreThen":"More than","registeredUsersPilotArea":"Join over {{ count }} registered users","registeredUsers":"{{ count }}\u003cbr\u003e registered users","registeredUsersPilot":"registered users","testimoni1":{"name":"Silje Glidseth","testimoni":"I am very pleased with the follow-up I received. This lead to new accomplishment and I have gotten better grades"},"testimoni2":{"name":"Thorbjørn Brook Steen","testimoni":"I sent my daughter to House of Math when she started middle school, not because she struggled with math, but because she needed motivation. Her teenage brain needed guidance as well as accelerated math education. We are extremely satisfied with both!"},"itWorks":"It works!"},"onboarding":{"completed":"completed","weHaveUpgraded":"We have upgraded the platform!","getACustomizedExperience":"Get a customized experience!","weHaveUpgradedSubtitle":"Get a new math task and a BIG bonus!","customizedSectionSubtitle":"Click the button to get the best possible math help!","retakeMappingQuiz":"Retake Mapping Quiz","tipsTooltipPrefix":"Click button to get your own","tipsTooltipSuffix":"Learning Path","customizeExperience":"Customize experience","notNow":"Not now","myAccountType":"Who are you?","buildMyLearningPath":"Build my Learning Path","buildYourPath":"Building your Learning Path","firstMathQuests":"Your first math tasks","student":"Student","teacher":"Teacher","parent":"Parent","other":"Other","next":"Next","skip":"Skip","myDetails":"My details","errors":{"nickname":"Nickname not provided","email":"Valid email not provided","noEmail":"E-mail not provided","emailTaken":"Email is already in use","birthYear":"Birth year not provided","country":"Country not provided","skill":"Math Level not provided"},"verifyEmail":"Verify the email","myMathLevel":"My math level","whatIsYourMathLevel":"What is your math level?","createYourOwnAvatar":"Create your own avatar","myAvatar":"My Avatar","makeAvatar":"Make avatar","exit":"Exit","myMappingQuiz":"My Mapping Quiz","forTheBestPossible":"For the best possible \u003cstrong\u003emath help\u003c/strong\u003e click the button","customize":"Customize","philosophy":{"pageOneTitle":"Hey you!","pageTwoTitle":"A smarter way is available","pageThreeTitle":"We are here to help","findPersonalized":"Find your personalized Learning Path on the next page","pageOneText":"Are you used to \u003cstrong\u003elearning math based on what grade\u003c/strong\u003e you are in?","pageTwoText":"Be prepared for \u003cstrong\u003eunexpected topics\u003c/strong\u003e to learn math smarter and easier","pageOneButton":"Yes","pageTwoButton":"Enter the unexpected","pageThreeButton":"Get your Learning Path"},"myLearningPath":"My Learning Path","toGiveCustom":"We need this so we can give you a custom experience","wantToDo":"What do you want to do?","findMathTopic":"Find a math topic","fillGaps":"Fill gaps","yourBirthYear":"What is the learner’s birth year?","addANickname":"Add a nickname","nickname":"Add nickname","continue":"Continue","birthYear":"Enter birth year..","algebra":"Algebra","geometry":"Geometry","numbersAndQuantities":"Numbers and Quantities","statisticsAndProbability":"Statistics and Probability","functions":"Functions","proofs":"Proofs","excellentChoice":"Excellent choice!","pickATopic":"Search or pick from the topics below","fewMoreSteps":"Just a few more steps","getStarted":"Help me get started","searchTip":"\u003cstrong\u003eTip:\u003c/strong\u003e Try searching in english if you can't find what you are looking for","curriculum":{"selectByGrade":"Select by grade","selectByTopic":"Select by Topic","toGetStarted":"To get started","selectYourGrade":"Select your grade","selectGrade":"Select Grade"},"intro":"Intro","brain":"brain","start":"Start","yourAssessmentQuiz":"Your Assessment Quiz!","mapYourMath":"We'll \u003cstrong\u003emap your math\u003c/strong\u003e and create a \u003cstrong\u003epersonalized learning path\u003c/strong\u003e.","optimizeYourLearning":"It adapts as you progress to \u003cstrong\u003eoptimize your learning\u003c/strong\u003e.","ok":"Оk"},"highscore":{"highscore":"Site score"},"emailVerify":{"title":"Verify your email","metaDescription":"Verify your email, or send a new verification email","resendEmail":"Resend verification link","sendEmail":"Send verification link","resend":"Resend","tooManyRequests":"You are trying this too often. Please try again later.","somethingWrong":"Could not resend email. Please try again later","notYourEmail":"Not your E-mail?","goToSettings":"Go to settings","expired":{"title":"Verification link has expired","text":"The verification link has expired. It expires 48 hours after you receive it","buttonLabel":"Click the button to send a new verification link"},"success":{"title":"Verification successful","text":"Congratulations, you have verified your email!"},"error":{"title":"UH-OH!","text":"Something went wrong","buttonLabel":"Try resending"},"resent":{"title":"A verification link has been sent to {{email}}","text":"Please click the link that has just been sent to","subTitle":"\u003cspan style='color: black'\u003eVerify your\u003c/span\u003e e-mail to learn math smarter and easier","verifyEmailText":"to verify your email","buttonLabel":"Did you not receive a link?"},"invalidEmail":"Your E-mail is not valid","notVerified":"E-mail not verified","rememberToVerify":"Verify your email and boost your math skills!","sendVerificationLink":"Verify my e-mail","checkSettings":"You can check if the e-mail is correct in the","emailVerificationLinkSent":"E-mail verification link sent","emailVerificationLinkFailedToSent":"Verification link failed to send","somethingWentWrong":"Something went wrong","addValidEmail":"Add a valid e-mail in","addAValidEmail":"Add a valid email","settings":"settings","tryAgain":"Send again","whatIsMyEmailText":"What e-mail am I registered with?","emailVerifyTooManyRequest":"Too many attempts wait 1 minute","emailDurationMessage":"This can take a few moments","emailVerifyAlt":"Email verification status is {{ status }}","exitIcon":"Exit icon","addEmail":"Add a valid e-mail","complete":"Complete","verify":"Verify","yourEmail":"Your email","emailVerifyMessage1":"Complete your profile to discover more ways to boost your math skills!","emailVerifyMessage2":"\u003cspan style='color: red'\u003eVerify\u003c/span\u003e your email to discover more ways to boost your math skills!","emailVerifyMessage3":"Complete your profile and \u003cspan style='color: red'\u003everify\u003c/span\u003e your email to discover more ways to boost your math skills!","pleaseCheck":"Please check","toVerify":"to verify your account","wantToChange":"Want to change your e-mail?","changeHere":"Change it here","toVerifyEmail":"to \u003cspan style='color: red'\u003everify\u003c/span\u003e your email"},"registerAccount":{"IamA":"I am a","birthYear":"Year of birth","button":"Next","chooseBestDesc":"Choose the alternative which most closely resembles you","country":"Country","details":"Fill in details","dropdownYearsInSchool":"Years in school","numberOfYearsInSchool":"Number of years in school","gradeLevel":"Grade level","haveUser":"Already have a user?","inMath":"In math I am","indAvatar":"Choose avatar","editAvatar":"You can edit your avatar later","pickAppearance":"Pick appearance","indBirth":"Birth year","indConfirm":"Confirm my account","indLevel":"Math level","myMathLevel":"My math level","indNickname":"My nickname","indRegister":"Sign up","indSchool":"School and grade level","indWhere":"Where are you from?","indWho":"Who are you?","initial":"Sign up","signUp":"Sign up","logIn":"Log in","logInForOther1":"If they already have a user, ","logInForOther2":"for them.","myInfo":"My information","nickname":"Choose a nickname","registerWithFacebook":"Continue with Facebook","registerWithGoogle":"Continue with Google","saveAndProceed":"Save and proceed","signUpForOther":"If you are signing up for someone else, use their information.","studentInfo":"Add student information","timezone":"Timezone","userSkill":"My math level is","userType":"I am","wantsNewsletter":"I would like relevant news and offers from houseofmath.com","workWith":"I want to work with","yearsAtSchool":"Number of years in school","detailLabel":"My details","accountType":"My account type","byContinuingYouAgreeToOur":"By continuing, you agree to our","and":"and","privacyPolicy":"Privacy Policy","termsAndConditions":"Terms and Conditions","continue":"Continue","extendedTerms":"This site is protected by reCAPTCHA. \u003cprivacyPolicy\u003eGoogle Privacy Policy\u003c/privacyPolicy\u003e and \u003ctermsAndConditions\u003eTerms and Conditions\u003c/termsAndConditions\u003e apply."},"adminPanel":{"allTime":"All time","assign":"Assign","assignTutor":"Assign tutor","assignRole":"Assign role","avgSizeNok":"Avg. package size (NOK)","avgSizeTotal":"Avg. package size Total","avgSizeUsd":"Avg. package size (USD)","cancelSession":"Cancel session","createZoomAccount":"Create zoom Account","earnings":"Earnings","earningsNok":"Revenue (NOK)","earningsUsd":"Revenue (USD)","endDate":"End date","hourEight":"8 hours T/Sr","hourEighty":"80 hours T/Sr","hourForty":"40 hours T/Sr","hourSixtyFour":"64 hours T/Sr","hourThirtyTwo":"32 hours T/Sr","hourFour":"4 hours T/Sr","hourSix":"6 hours T/Sr","hourSixteen":"16 hours T/Sr","hours":"Hours","inThousand":"in 1000","international":"International","latestBooking":"Latest booking","location":"Location","tutorStat":"Tutor on Demand stats","selectSubscriptionPackage":"Select Stripe Package","selectStripeProduct":"1. Select Stripe product","selectStripePrice":"2. Select pricing","selectCountryLabel":"3. Select country","freeTrialLabel":"4. Free Trial","selectCountry":"Select Country","stripeProducts":"Stripe Products","stripeProduct":"Stripe product","paymentGateway":"Payment Gateway","priceApiId":"Price API ID","price":"Price","pricing":"Pricing","country":"Country","city":"City","region":"Region","zip":"Zip","lat":"Lat","lng":"Lng","radius":"Radius","default":"Default","deleteCountryAssociation":"Delete country association","subscription":"Subscription","add":"Add","noRecordFound":"No record found","stripeProductArchived":"Product has been archived","stripePriceArchived":"Price has been archived","norwegian":"Norwegian","numberOfChatMessages":"Chat messages: {{ messages }}","numberOfCountries":"Country Count: {{ countries }}","numberOfTutors":"Tutor Count: {{ tutors }}","existingStudents":"Existing students: {{ students }}","numberOfNewStudents":"New students: {{ students }}","totalNumberOfStudents":"Total number of students: {{ students }}","numberOfUsers":"User Count: {{ users }}","overview":"Overview","packageType":"Package type","packages":"Packages\t","recommended":"Recommended","scheduleFilter":{"available":"Available","chat":"Chat","on-duty":"On duty","selectAvailabilityType":"Select availability type","selectPurpose":"Select purpose","video-chat":"Video Chat"},"searchTutors":"Search Tutors","searchStudents":"Search Students","searchUsers":"Search Users","seeDetails":"See Details","selectRole":"Select Role","sessions":"Sessions","singleHour":"Single Hour T/Sr","startDate":"Start date","teacherMessages":"From teachers","trialPeriodDays":"Trial Period Days","title":"Admin Panel","total":"Total T/Sr","totalBought":"Total hours bought","totalUsed":"Total hours used","transferSession":"Transfer session","unassign":"Unassign","unassignTutor":"Unassign tutor","userMessages":"From users","allCustomers":"All customers","campaignCustomers":"Campaign customers","userTypes":{"enthusiast":"Enthusiast","grandparent":"Grandparent","null":"Not chosen","other":"Other","parent":"Parent","pupil":"Pupil","student":"Student","teacher":"Teacher"},"videoChat":"Video chat","studentSettings":{"saveLabel":"Save","updateSuccessful":"Update successful"},"userAssignRoles":{"freeUser":"Free user","user":"User","teacher":"Teacher","mentor":"Tutor","admin":"Admin","student":"Student","videoChatMentor":"Video chat tutor"},"userPermissions":{"tod_admin":"TOD Admin","subscription_admin":"Subscription Admin","tutor_manager":"Tutor Manager","chat_supervisor":"Chat Supervisor","statistics_read":"Statistics Viewer","selectPermissions":"Select Permissions","permissionFor":"Set permissions for","permissionBtnText":"Set Permissions","somethingWentWrong":"Something went wrong"},"authEmail":"Oauth E-mail","authProvider":"Oauth Provider","permissions":"Permissions","permissionAddText":"Add new subscription permissions","feature":"Feature","selectFeature":"Select Feature","urlExtension":"URL Extension","timerBasedAppear":"Timer-based appearance","timerBased":"Timer based","urlExtensionPlaceholder":"/* or /topic or /topic/sub-topic","subscriptionPermissions":"Subscription permissions","timeOpen":"Time Open","timeOpenAddText":"Add time spent for feature","subscriptionTimeOpen":"Subscription Time Open","editTimeOpen":"Edit Time Open","minutes":"Minutes","seconds":"Seconds","delete":"Delete","remove":"Remove","cancel":"Cancel","save":"Save","edit":"Edit","from":"from","scheduleSession":{"addTutorSession":"Add session for tutor {{ tutorName }}"},"youHaveUnsavedChanges":"You have unsaved changes","saveChanges":"Save changes","informationUpdated":"Information Updated","toolTip":"Manage your subscription and tutor on demand."},"tutorPanel":{"closeCalendar":"Close calendar","myCalendar":"My calendar","nextAppointment":"Next appointment","scheduleSession":{"settingForTutor":"Adding session for tutor with id {{ tutorId }}","addInfo":"Add info","addTime":"Add time","buttonConfirm":"Confirm","buttonNext":"Next","date":"Date","errorTime":"End time must be after start time.","minTimeErr":"Minimum Session Time should be 1 hour","goBack":"Back to overview","info":"Info","selectDate":"Select date","time":"Time","errors":{"overbook_student":"Cannot overbook student","same_student":"Sessions are not for the same student","one_year_valid":"Booking is valid for one year","mentor_not_assigned":"Tutor is not assigned to booking","timeslot_conflict":"Session timeslots are conflicting"}},"degree":{"associate":"Associate Degree","bachelor":"Bachelor's Degree","master":"Master's Degree","doctoral":"Doctoral Degree","risingStar":"Rising star"},"seeDetails":"See details","sessionComment":{"commentUpdated":"Comment was updated.","validationError":"Comment must be between 5 and 255 characters."},"settings":{"labelBio":"Small description about you","labelCity":"City","labelEducationalInstitution":"Educational institution","labelEducationalProgramme":"Programme","labelEducationalDegree":"Educational degree","labelFullName":"Full name","labelGradeLevels":"Grades I teach","labelLanguages":"Language I tutor","labelLocation":"Location","labelPhoneNumber":"Phone number","labelPreferredDays":"Available tutoring times","labelSelectCity":"Location I tutor","labelSelectLanguage":"Select language","labelSelectTimezone":"Select timezone","labelSubjects":"Subjects I teach","labelTimezone":"Timezone","labelTraits":"My personality traits","labelVideoChatAvailability":"Video Chat availability","descriptionLevelOfEducation":"Choose the grade levels you can teach","descriptionSubjects":"Choose the subjects you can teach","descriptionGradeLevels":"For Norwegian school system. Select the math topics you are able to teach"},"studentCard":{"buttonScheduleSession":"Schedule next session","email":"E-mail","lastSession":"Last session","minutesLeft":"Minutes left","name":"Name","newStudent":"New student - Schedule session!","nextSession":"Next session","numSessions":"Amount of hours completed","phoneNumber":"Phone number","timezone":"Timezone"},"videoChat":{"todaysTasks":"Today's tasks"},"title":"Tutor panel","welcome":"Welcome","tutor":"Tutor","noTaskToday":"No task today","toolTip":"Manage your tutor sessions."},"studentPanel":{"bookAVideoChatSession":"Book a video chat session","bookNewVideoChat":"Book a 15 min video chat","buyPackage":{"hours_8":"Buy 8 hours package","hours_16":"Buy 16 hours package","latest":"Buy latest package"},"purchaseHistory":"Purchase history","buttonBecomeAStudent":"Book tutoring","completePreferencesButton":"Complete registration","completePreferencesHeader":"Tutor pending: Complete registration","completePreferencesText":"In order to find a suitable tutor, please complete the registration","hours":"Hours","tutorContactInfo":"Tutor contact info","tutorPending":"Tutor pending","tutorPendingMessage":"Your tutor will show up here as soon as possible.","metaDescription":"The student panel gives you a complete overview of your private tutoring. Use the dashboard to communicate with your personal tutor from House of Math.","myPackages":"My packages","newBooking":"New booking","nextAppointment":"Next appointment","noCompletedSessionsYet":"You have no completed sessions yet.","observingUser":"Observing {{ user }}","ratingHeader":"How was your session experience with {{ name }} on {{ date }}?","resendSignUpEmail":"Resend sign-up email","sessionsCompleted":"Sessions completed","settings":{"descriptionGradeLevels":"Since you have chosen the Norwegian school system, grades 11 to 13, and math as a subject, you need to specify which grade levels within math you want to focus on.","descriptionLevelOfEducation":"Choose how many years you have attended school","descriptionSubjects":"Choose the subjects you want to focus on.","descriptionTraits":"Choose up to three preferred tutor personality traits.","labelFullName":"Full name","labelGradeLevels":"Grade levels","labelLanguages":"Languages","labelLocation":"Location","labelPhoneNumber":"Phone number","labelPreferredDays":"Preferred days","labelSchoolCountry":"Where do you go to school?","labelSelectCity":"Select city","labelSelectLanguage":"Select language","labelSubjects":"Subjects","labelTimezone":"Timezone","labelTraits":"Tutor personality traits"},"studentDetails":"Student Details","title":"Student panel","userNeedsToCreateAccount":"\u003cstrong\u003e{{ email }}\u003c/strong\u003e needs to create an account in order for us to provide the package.","toolTip":"Find information on your appointments, tutor and receipts."},"statistics":{"pageTitle":"Progress {{username}}","statistics":"Progress","metaDescription":"See how you are progressing in different math topics, and keep track of all the rewards you have earned!","productCategoryFilters":{"all":"All Math Categories","algebra":"Algebra","geometry":"Geometry","numbers-and-quantities":"Numbers and Quantities","statistics-and-probability":"Statistics and Probability","functions":"Functions","proofs":"Proofs"},"learning":"Learning","rewards":"Rewards","all-time":"All Time","last-7-days":"Last 7 days","last-30-days":"Last 30 days","last-90-days":"Last 90 days","last-year":"Last year","video-course":"Video Course","math-vault":"Junior Math","last-visited":"Last visited","today":"Today","yesterday":"Yesterday","activities":"Activities","activities-done":"Activities done","activity-accuracy":"Activity accuracy","total-time-spent":"Total time spent","video-watching-time":"Video watching time","xp-earned":"XP earned","kudos-earned":"Kudos earned","gems":"Gems","fun-fact-chests":"Fun fact chests","correct-answer":"Correct answers","activity-sets":"Activity sets","videos":"Videos","level":"Level","levels":"Levels","boss-battle":"Boss battle","continue":"Continue","play":"Play","set":"Set {{n}}","correct":"Correct","got-to-video-courses":"Go to Bootcamps","you-have-not-visited":"You have not visited any learning areas the last {{n}} days","you-have-not-visited-all-time":"You have not visited any learning areas","create-nickname":"Create a nickname","answered":"Answered {{n}}","go-to-statistics":"My stats","review":"Review","timestat":{"days":"d","hours":"h","minutes":"min"},"timeStamps":{"days":"{{days}} d","hours":"{{hours}} h","minutes":"{{minutes}} min"},"chart":{"average":"Average","minutes":"min","per-day":"per day","dialog":"Build math skills in 15 min a day!","noDataText1":"Do math 15 min a day to get","noDataText2":"a better grade","week":"Week"},"last_attempts":"Last attempts","attempts":"Attempts","do_activities_to_see_your_attempts":"Complete activity sets to see your attempts","alt_texts":{"chart":"History of attempts","retake":"Retake","go":"Go","vcc":"Bootcamps icon","activities_done":"Sheet with exercises that have been graded","activity_accuracy":"Archery target","clock":"Clock","video_camera":"Video camera","xp":"XP","kudos":"Kudos","gems":"Gems","chest":"Chest","answered":"Sheet with exercises that have been graded","completed":"Completed","math_essential":"Math Essential Icon","article":"Article Icon","tick":"Tick Icon","book":"Book Icon","league":"League Icon"},"re_do_activities_to_see_your_attempts":"Since your last attempt we've updated the activity sets. Future attempts will show here.","math_essential":"Math Essential","quizzes":"Quizzes","quiz":"Quiz","articles":"Articles","math-essentials-subjects":{"the-multiplication-table":"Multiplication","fractions":"Fractions","percent":"Percentages","powers":"Powers, Roots and Scientific Notation","algebraic-factorization":"Algebraic Expressions","basic-algebra":"Algebraic Expressions","parentheses":"Algebraic Expressions","equations":"Equations","functions":"Functions"},"fractions":"Fractions","game":"Game","stars":"Stars","gold_boxes":"Gold boxes","highscore":"Highscore","stage_reached":"Stage reached","stage":"Stage","levels_completed":"Levels completed","windSurfer":{"bestScore":"Best Score","bestStreak":"Best Streak","bestLevel":"Best Level","run":"Run","score":"Score","last-five":"Last 5 Runs"},"multGame":{"multiplication":"Multiplication","solved":"Multiplications solved","accuracy":"Accuracy","bestStreak":"Best streak","score":"Score","survival":"Survival","sprint":"100 Sprint","table-master":"Table Master"},"video-crash-courses":"Bootcamps","math-essentials":"Math Essentials","games":"Games","highestLeague":"Highest all-time league","statisticsCardButton":"Progress","playGames":"Games","toolTip":"View your learning progress and rewards.","mathArticle":"Math Article","articleRead":"Article read"},"avatar":{"acceptOrSkip":"Accept terms and conditions for using avatar","affirm":"You are using WOLF3D's avatar creator.","avatarDescChoose":"Pick appearance","avatarTitle":"Create an avatar","avatarTitleChoose":"Choose an avatar","consent":"I'm 13 years or older and have read the terms and conditions for using avatars","createAvatarButton":"Make avatar","createAvatarMultGame":"Create your own avatar to use in Multiplication Master","creatingAvatar":"Only a few seconds now \u003cbr\u003e We are creating your avatar","currentLeague":"Current league","editAvatar":"Edit avatar","feminine":"Girl","makeAvatar":"Create an avatar","masculine":"Boy","nextPage":"Next","skipPage":"Skip","terms":"An avatar is made through our partner (Wolf3D). We would like to inform you that they have their own terms and conditions which must be accepted to make an avatar. We are not responsible for the content of these terms and conditions, the content of their website or for personal information gathered through their website when you create an avatar. We ask you therefore to examine their terms and conditions before you submit personal data.","termsCheck":"I'm 13 years or older and have read the terms and conditions for using avatars","termsTitle":"Terms and conditions for using avatars","under13":"I’m under 13","userTerms":"User terms","waiting":"Please wait...","wolf3DLogo":"Wolf3D logo","yourAvatar":"Your avatar","outdatedAvatar":"Utdatert avatar","rewards":"Rewards!","congratulations":"Congratulations!","feedback":{"great":"Great!","awesome":"Awesome!","cool":"Cool!","nice":"Nice!","impressive":"Impressive!","amazing":"Amazing!"}},"bootcampQuiz":{"quizPage":"Boss Battle","start_quiz":"Start Boss Battle","next":"Next","multiplication_numbers-and-quantities_instruction_title":"Instruction","multiplication_numbers-and-quantities_instruction_description":"Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.","skip":"Skip","submit":"Submit","check":"Check","ok":"Ok!","try_again!":"Try again!","you_earned":"You earned","no_of_correct_answers":"Number of correct answers","victory":"Victory","defeat":"Defeat","health":"Boss Health","boss":"Boss","bossBackground":"Boss Background","player":"Player","vs":"vs","TryDefeatBoss":"Try again and defeat the boss"},"bootcampActivity":{"geometryActivities":{"line":"Line","ray":"Ray","lineSegment":"Line segment","edge":"Edge","vertex":"Vertex","angle":"Angle","acute":"Acute","right":"Right","obtuse":"Obtuse","triangle":"Triangle","quadrilateral":"Quadrilateral","pentagon":"Pentagon","hexagon":"Hexagon","heptagon":"Heptagon","octagon":"Octagon","yes":"Yes","no":"No"}},"panels":{"tab":{"academy":"HOM Academy","appointments":"Appointments","calendar":"Calendar","chat":"Chat","home":"Home","tutors":"Tutors","menu":"Menu","myTutor":"My tutor","myStudents":"My students","observer":"Observer mode","receipts":"Receipts","schedule":"Schedule","settings":"Settings","statistics":"Statistics","students":"ToD","users":"Users","videoChat":"Video chat","stripe":"Stripe","permissions":"Permissions","featureFreeAccessDuration":"Time Open"}},"leaderboard":{"title":"Leagues","background":"Leaderboard background","avatar":"User avatar","metaDescription":"Use the House of Math leaderboard to track your progress, compete with your friends, and elevate your math learning experience! Start having fun while following your math journey.","resetIn":"Leaderboard will reset in","currentRank":"Your current rank","rank":"Rank","nickname":"Nickname","XPtoLead":"XP to lead","joinLeaderboard":"Earn XP to join this week’s league","unranked":"Currently unranked","globalLeaderboard":"Global Leaderboard","weekNo":"Week {{ weekNo }}","compareToOther":"See how you compare to other users","newBoardIn":"New weekly leaderboard in:","days":"d","hours":"h","minutes":"min","top100":"Top 100","current":"Current","lastWeek":"Last week","yourRank":"Your rank","top3":"Top 3","lastWeeksBoard":"Last Week's","toLearningPath":"Go to Learning Tracks","league":{"unranked":"Unranked","bronze":"Bronze League","silver":"Silver League","gold":"Gold League","platinum":"Platinum League","diamond":"Diamond League"},"badges":{"unranked":"Unranked","bronze":"Bronze","silver":"Silver","gold":"Gold","platinum":"Platinum","diamond":"Diamond"},"topAdvances":"Top {{nrOfPromotions}} advances in:","secureSpotInTop":"Stay in the Top {{nrOfStays}} to avoid demotion!","leaguePromotion":"League Promotion","leagueDemotion":"League Demotion","waitingForPlayers":"Waiting for players:","LeagueNotFull":"League not filled","needMorePeople":"You can be promoted or demoted in the league after {{leaderboardCapacity}} players have joined.","notParticipating":"Not Participating","enableLeague":"Click enable league participating if you want to compete in the league","littleMore":"Just a little more XP to move up!","keepGoing":"You're on fire! Keep going!","timeEnds":"Period ends in: \u003cspan\u003e{{time}}\u003c/span\u003e","xpToNext":"\u003cspan\u003e{{xp}} XP\u003c/span\u003e to next rank!"},"encyclopedia":{"back":"Go back","buttonQuizYourself":"Quiz yourself","customizedView":"Customized View","customizedViewTooltip":"Customized View shows you entries for the level you chose in settings. Toggle to see all entries.","filter":"Filter","introMessage":"This encyclopedia contains numerous entries of mathematics. Search for entries using the search field or navigate by using the menu.","nextEntry":"Next entry","prevEntry":"Previous entry","quizAtTheBottomMessage":"At the bottom of the page there is an interactive quiz.","readHere":"Read here","titleExcel":"Excel","titleGeogebra":"GeoGebra","overviewAndExamTipsIcon":"Overview and Exam Tips Icon","economyIcon":"Economy icon","statisticsIcon":"Statistics icon","numbersAndQuantitiesIcon":"Numbers and Quantities icon","algebraIcon":"Algebra icon","geometryIcon":"Geometry icon","statisticsAndProbabilityIcon":"Statistics and Probability icon","functionsIcon":"Functions icon","proofsIcon":"Proofs icon","previewAltPrefix":"Article on {{ title }}","previewAltOGP":"Preview of {{ title }}","knowMore":"Want to know more?","singUP":"Sign Up","itsFree":"It's free!"}}},"initialLocale":"en","ns":["imageAlt","curriculum","footer","generics","errors","campaign","common","bootcamp","subscription","chat","chatAi","userFeedback","chat","gemShop","homebase","menu","navbar","notFound","search","dashboard","gamification","home","onboarding","highscore","emailVerify","registerAccount","subscription","imageAlt","adminPanel","tutorPanel","studentPanel","statistics","home","avatar","bootcamp","bootcampQuiz","bootcampActivity","panels","leaderboard","encyclopedia"],"userConfig":{"i18n":{"defaultLocale":"en","locales":["en","no","uk"]},"localePath":"/var/task/public/locales","reloadOnPrerender":false,"default":{"i18n":{"defaultLocale":"en","locales":["en","no","uk"]},"localePath":"/var/task/public/locales","reloadOnPrerender":false}}},"initialState":"{\"user\":{\"current\":null,\"cookieConsentApplicable\":false},\"adminSchedule\":{\"selectedTimeslotSchedules\":[]},\"auth\":{\"jwtPayload\":null,\"firebaseAuthenticated\":false},\"avatar\":{\"avatarFields\":{\"avatarUrl\":\"\",\"avatarPng\":\"\",\"childAvatar\":false},\"avatarError\":false,\"downloadUrl\":\"\",\"avatarPng\":\"\"},\"subscription\":{\"subscriptionSuccessRedirectUrl\":\"\",\"subscriptionProduct\":{},\"currentSubscriptionPriceId\":\"\",\"currentSubscriptionId\":\"\",\"allowTrial\":false,\"isGa4Logged\":false,\"isSubscriptionApplicable\":false,\"showSubscriptionPackages\":false},\"bootcampActivity\":{\"correctConsecutiveAnswers\":0,\"currActivityType\":1,\"currDifficultyLevel\":1,\"currRewardLevel\":1,\"answerState\":null,\"didUpdateEvent\":false,\"events\":[],\"currentRound\":0,\"shouldDisableUserInteraction\":false,\"shouldGetEnergyOrbs\":false,\"shouldGetActivityHint\":false,\"shouldAutoValidateAnswer\":false,\"collectedRewards\":0,\"kudos\":0},\"bootcampPage\":{\"activities\":[],\"chest\":{},\"content\":[],\"help\":{\"encyclopediaPath\":\"\",\"drillPath\":\"\"},\"isCompleted\":false,\"sectionName\":\"\",\"usefulWords\":[],\"isCollapsed\":false,\"encyclopediaArticle\":{\"breadcrumbs\":[{\"_key\":\"bcedd644-0bec-506b-86c8-f5dfe20c2b99\",\"h1\":\"GeoGebra\",\"path\":\"/geogebra\"},{\"_key\":\"ef61a895-ac2e-50db-8973-568345df9c6a\",\"h1\":\"Numbers and Quantities\",\"path\":\"/geogebra/numbers-and-quantities\"},{\"_key\":\"e0c1b714-a308-5187-9ae8-da563b5bd966\",\"h1\":\"Vectors\",\"path\":\"/geogebra/numbers-and-quantities/vectors\"},{\"_key\":\"225d9721-9be2-52ce-bdde-c3e13b3fcef0\",\"h1\":\"Three Dimensions\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions\"},{\"_key\":\"ff5fd763-7e2e-512a-9ec2-5fe3a9052078\",\"h1\":\"Find Angles Between Objects in Space with GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/find-angles-between-objects-in-space-with-geogebra\"}],\"cssUrl\":\"/0e751716-e4cf-4658-ac87-c97d9167170e-en-2.css\",\"h1\":\"Find Angles Between Objects in Space with GeoGebra\",\"htmlContent\":{\"html\":\" \u003c/p\u003e\u003cp class=\\\"indent\\\" \u003e     You can find the angles between two \u003ca href=\\\"/encyclopedia/numbers-and-quantities/vectors/three-dimensions/parametrization/parameterization-of-lines-in-space\\\"\u003elines\u003c/a\u003e in 3D space, a line and a \u003ca href=\\\"run:encyclopedia/Planes\\\"\u003eplane\u003c/a\u003e and two planes in 3D space by using \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e. You can also read the entry about \u003ca href=\\\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-make-lines-and-planes-in-geogebra\\\"\u003edrawing lines and planes in 3D space\u003c/a type=\\\"punct\\\"\u003e. \u003c/p\u003e      \u003cdiv class=\\\"tcolorbox foogeogebra\\\" id=\\\"tcolobox-563\\\"\u003e      \u003cdiv class=\\\"tcolorbox-title\\\"\u003e                                                                                                                                                                                                                                          \u003cp class=\\\"indent\\\" \u003e     \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e Instruction 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\\\"tcolorbox-content\\\"\u003e\u003cp class=\\\"noindent\\\" \u003e\u003ch3 class=\\\"geogebra-container-title\\\"\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eFinding\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ethe\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eAngle\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eBetween\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eTwo\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eLines\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\\\"enumerate-enumitem\\\"\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         Open         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     and         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e3D\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGraphics\u003c/span\u003e\u003c/code\u003e     under         \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     in         \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eMenu\u003c/span\u003e\u003c/code type=\\\"punct\\\"\u003e.         \u003c/dd\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         In         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e,         type         in         the         expressions         of         the         two         lines         that         are         crossing         each         other         in         3D         space.         It         would         be         wise         to         make         sure         they         have         a         point         of         intersection,         so         you         know         for         certain         that         they         \u003cspan  class=\\\"ec-lmri-12x-x-144\\\"\u003ewill\u003c/span\u003e        cross         each         other.         \u003c/dd\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003eIn \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e, type         \u003cdiv class=\\\"flushleft\\\"  \u003e \u003cp class=\\\"noindent\\\" \u003e   \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAngle(\u0026#x003C;line\u0026#x003E;, \u0026#x003C;line\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e     \u003cp class=\\\"noindent\\\" \u003ewhere you enter the name of each line in each of the \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e\u0026#x003C;line\u0026#x003E;\u003c/span\u003e\u003c/code\u003e fields. \u003c/p\u003e\u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                         \u003cdiv class=\\\"tcolorbox fooqaexample\\\" id=\\\"tcolobox-564\\\"\u003e      \u003cdiv class=\\\"tcolorbox-title\\\"\u003e \u003cp class=\\\"indent\\\" \u003e     Example 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\\\"tcolorbox-content\\\"\u003e\u003cp class=\\\"noindent\\\" \u003e\u003cb\u003e\u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eFind the angle between the lines \u003c/span\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eand \u003c/span\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003ewith the parametrizations\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"block\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmtable  displaystyle=\\\"true\\\" columnalign=\\\"left\\\" class=\\\"align-star\\\"\u003e                              \u003cmtr\u003e\u003cmtd  columnalign=\\\"right\\\" class=\\\"align-odd\\\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\\\"align-even\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\" type=\\\"divide\\\"\u003e:\u003c/mo\u003e \u003cmo  class=\\\"MathClass-open\\\" stretchy=\\\"false\\\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\\\"MathClass-close\\\" stretchy=\\\"false\\\"\u003e)\u003c/mo\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmi  \u003et\u003c/mi\u003e\u003cmspace width=\\\"-0.17em\\\" class=\\\"negthinspace\\\"/\u003e \u003cmrow\u003e\u003cmo fence=\\\"true\\\" form=\\\"prefix\\\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\\\"true\\\" form=\\\"postfix\\\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\\\"right\\\" class=\\\"align-label\\\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\\\"align-label\\\"\u003e                              \u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\\\"right\\\" class=\\\"align-odd\\\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\\\"align-even\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eb\u003c/mi\u003e \u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\" type=\\\"divide\\\"\u003e:\u003c/mo\u003e \u003cmo  class=\\\"MathClass-open\\\" stretchy=\\\"false\\\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\\\"MathClass-close\\\" stretchy=\\\"false\\\"\u003e)\u003c/mo\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmi  \u003et\u003c/mi\u003e\u003cmspace width=\\\"-0.17em\\\" class=\\\"negthinspace\\\"/\u003e \u003cmrow\u003e\u003cmo fence=\\\"true\\\" form=\\\"prefix\\\"\u003e(\u003c/mo\u003e\u003cmrow\u003e\u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e−\u003c/mo\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e\u003cmo fence=\\\"true\\\" form=\\\"postfix\\\"\u003e)\u003c/mo\u003e\u003c/mrow\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\\\"right\\\" class=\\\"align-label\\\"\u003e\u003c/mtd\u003e                             \u003cmtd  class=\\\"align-label\\\"\u003e \u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\\\"noindent\\\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\\\"lowerbox\\\"\u003e                                                                                                                                                                                                                                           \u003cp class=\\\"noindent\\\" \u003eFirst, you type the expressions \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math\u003e into \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e. If you wish to have an illustration of the lines, you can open \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e3D Graphics\u003c/span\u003e\u003c/code\u003e under \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e in \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eMenu\u003c/span\u003e\u003c/code\u003e. Then use \u003c/p\u003e \u003cdiv class=\\\"flushleft\\\"  \u003e \u003cp class=\\\"noindent\\\" \u003e   \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAngle(\u0026#x003C;line\u0026#x003E;, \u0026#x003C;line\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e \u003cp class=\\\"noindent\\\" \u003eto find the angle between \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eb\u003c/mi\u003e\u003c/math type=\\\"punct\\\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e      \u003cdiv class=\\\"tcolorbox foogeogebra\\\" id=\\\"tcolobox-565\\\"\u003e      \u003cdiv class=\\\"tcolorbox-title\\\"\u003e                                                                                                                                                                                                                                          \u003cp class=\\\"indent\\\" \u003e     \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e Instruction 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\\\"tcolorbox-content\\\"\u003e\u003cp class=\\\"noindent\\\" \u003e\u003ch3 class=\\\"geogebra-container-title\\\"\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eFinding\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ethe\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eAngle\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eBetween\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ea\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eLine\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eand\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ea\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ePlane\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\\\"enumerate-enumitem\\\"\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         Open         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     and         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e3D\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGraphics\u003c/span\u003e\u003c/code\u003e     under         \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     in         \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eMenu\u003c/span\u003e\u003c/code type=\\\"punct\\\"\u003e.         \u003c/dd\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         In         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     type         in         the         expressions         of         your         line         and         plane         that         are         crossing         each         other.         It         would         be         wise         to         make         sure         they         have         a         point         of         intersection,         so         that         they         will         cross         each         other.         \u003c/dd\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003eIn \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e type in         \u003cdiv class=\\\"flushleft\\\"  \u003e \u003cp class=\\\"noindent\\\" \u003e   \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAngle(\u0026#x003C;line\u0026#x003E;, \u0026#x003C;plane\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e     \u003cp class=\\\"noindent\\\" \u003ewhere you type in the name of the line in the \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e\u0026#x003C;line\u0026#x003E;\u003c/span\u003e\u003c/code\u003e field and the name of the         plane in the \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e\u0026#x003C;plane\u0026#x003E;\u003c/span\u003e\u003c/code\u003e field. \u003c/p\u003e\u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                         \u003cdiv class=\\\"tcolorbox fooqaexample\\\" id=\\\"tcolobox-566\\\"\u003e      \u003cdiv class=\\\"tcolorbox-title\\\"\u003e \u003cp class=\\\"indent\\\" \u003e     Example 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\\\"tcolorbox-content\\\"\u003e\u003cp class=\\\"noindent\\\" \u003e\u003cb\u003e\u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eFind the angle between the line \u003c/span\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eand the plane \u003c/span\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e\u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003e,\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003ewhere\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"block\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmtable  displaystyle=\\\"true\\\" columnalign=\\\"left\\\" class=\\\"align-star\\\"\u003e                               \u003cmtr\u003e\u003cmtd  columnalign=\\\"right\\\" class=\\\"align-odd\\\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\\\"align-even\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003ea\u003c/mi\u003e \u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\" type=\\\"divide\\\"\u003e:\u003c/mo\u003e \u003cmo  class=\\\"MathClass-open\\\" stretchy=\\\"false\\\"\u003e(\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\\\"MathClass-close\\\" stretchy=\\\"false\\\"\u003e)\u003c/mo\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmi  \u003et\u003c/mi\u003e\u003cmo  class=\\\"MathClass-open\\\" stretchy=\\\"false\\\"\u003e[\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\"\u003e,\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\\\"MathClass-close\\\" stretchy=\\\"false\\\"\u003e]\u003c/mo\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\\\"right\\\" class=\\\"align-label\\\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\\\"align-label\\\"\u003e                               \u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\\\"right\\\" class=\\\"align-odd\\\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\\\"align-even\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e \u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\" type=\\\"divide\\\"\u003e:\u003c/mo\u003e \u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\\\"MathClass-rel\\\" stretchy=\\\"false\\\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\\\"right\\\" class=\\\"align-label\\\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\\\"align-label\\\"\u003e \u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\\\"noindent\\\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\\\"lowerbox\\\"\u003e                                                                                                                                                                                                                                           \u003cp class=\\\"noindent\\\" \u003eYou type in the expressions of \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math\u003e in \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e. If you wish to have an illustration, you can open \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e3D Graphics\u003c/span\u003e\u003c/code\u003e under \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e in \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eMenu\u003c/span\u003e\u003c/code\u003e. Then use  \u003c/p\u003e \u003cdiv class=\\\"flushleft\\\"  \u003e \u003cp class=\\\"noindent\\\" \u003e   \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAngle(\u0026#x003C;Line\u0026#x003E;, \u0026#x003C;Plane\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e \u003cp class=\\\"noindent\\\" \u003eto find the angle between the line \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e and the plane \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math type=\\\"punct\\\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e      \u003cdiv class=\\\"tcolorbox foogeogebra\\\" id=\\\"tcolobox-567\\\"\u003e      \u003cdiv class=\\\"tcolorbox-title\\\"\u003e                                                                                                                                                                                                                                          \u003cp class=\\\"indent\\\" \u003e     \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGeoGebra\u003c/span\u003e\u003c/code\u003e Instruction 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\\\"tcolorbox-content\\\"\u003e\u003cp class=\\\"noindent\\\" \u003e\u003ch3 class=\\\"geogebra-container-title\\\"\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eFinding\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ethe\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eAngle\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eBetween\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003eTwo\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-172\\\"\u003ePlanes\u003c/span\u003e \u003c/h3\u003e      \u003c/p\u003e\u003cdl class=\\\"enumerate-enumitem\\\"\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e    1.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         Open         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     and         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e3D\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eGraphics\u003c/span\u003e\u003c/code\u003e     under         \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     in         \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eMenu\u003c/span\u003e\u003c/code type=\\\"punct\\\"\u003e.         \u003c/dd\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e     2.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         In         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     type         in         the         expressions         for         two         intersecting         place.         Make         sure         they         actually         intersect.         \u003c/dd\u003e\u003cdt class=\\\"enumerate-enumitem\\\"\u003e     3.  \u003c/dt\u003e\u003cdd  class=\\\"enumerate-enumitem\\\"\u003e         In         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e     type         in         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAngle(\u0026#x003C;plane\u0026#x003E;,\u003c/span\u003e     \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e\u0026#x003C;plane\u0026#x003E;)\u003c/span\u003e\u003c/code\u003e     where         you         type         in         the         names         of         the         planes         in         the         \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e\u0026#x003C;plane\u0026#x003E;\u003c/span\u003e\u003c/code\u003e     fields.         \u003c/dd\u003e\u003c/dl\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                         \u003cdiv class=\\\"tcolorbox fooqaexample\\\" id=\\\"tcolobox-568\\\"\u003e      \u003cdiv class=\\\"tcolorbox-title\\\"\u003e \u003cp class=\\\"indent\\\" \u003e     Example 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\\\"tcolorbox-content\\\"\u003e\u003cp class=\\\"noindent\\\" \u003e\u003cb\u003e\u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eFind the angle between the plane\u003c/span\u003e \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eα\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eand the\u003c/span\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003eplane \u003c/span\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003cspan  class=\\\"ec-lmbx-12x-x-144\\\"\u003egiven by\u003c/span\u003e \u003c/p\u003e\u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"block\\\" mathvariant=\\\"bold-italic\\\"\u003e\u003cmtable  displaystyle=\\\"true\\\" columnalign=\\\"left\\\" class=\\\"align-star\\\"\u003e                        \u003cmtr\u003e\u003cmtd  columnalign=\\\"right\\\" class=\\\"align-odd\\\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\\\"align-even\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eα\u003c/mi\u003e \u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\" type=\\\"divide\\\"\u003e:\u003c/mo\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e−\u003c/mo\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmi  \u003ex\u003c/mi\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmn\u003e1\u003c/mn\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmn\u003e7\u003c/mn\u003e\u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\\\"MathClass-rel\\\" stretchy=\\\"false\\\"\u003e=\u003c/mo\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e−\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e \u003cmo  class=\\\"MathClass-rel\\\" stretchy=\\\"false\\\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\\\"right\\\" class=\\\"align-label\\\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\\\"align-label\\\"\u003e                        \u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\\\"right\\\" class=\\\"align-odd\\\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\\\"align-even\\\"\u003e\u003cmstyle mathvariant=\\\"bold-italic\\\"\u003e\u003cmstyle mathvariant=\\\"bold\\\"\u003e\u003cmi  \u003eβ\u003c/mi\u003e \u003cmo  class=\\\"MathClass-punc\\\" stretchy=\\\"false\\\" type=\\\"divide\\\"\u003e:\u003c/mo\u003e \u003cmi  \u003ey\u003c/mi\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e+\u003c/mo\u003e \u003cmi  \u003ez\u003c/mi\u003e \u003cmo  class=\\\"MathClass-bin\\\" stretchy=\\\"false\\\"\u003e−\u003c/mo\u003e \u003cmn\u003e5\u003c/mn\u003e \u003cmo  class=\\\"MathClass-rel\\\" stretchy=\\\"false\\\"\u003e=\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\\\"right\\\" class=\\\"align-label\\\"\u003e\u003c/mtd\u003e                       \u003cmtd  class=\\\"align-label\\\"\u003e \u003cmspace width=\\\"2em\\\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e \u003cp class=\\\"noindent\\\" \u003e\u003c/b\u003e                                                                                          \u003cdiv class=\\\"lowerbox\\\"\u003e                                                                                                                                                                                                                                           \u003cp class=\\\"noindent\\\" \u003eYou type in the expressions of \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eα\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math\u003e in \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAlgebra View\u003c/span\u003e\u003c/code\u003e. If you wish to have an illustration, you can open \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003e3D Graphics\u003c/span\u003e\u003c/code\u003e under \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-view-bar.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eView\u003c/span\u003e\u003c/code\u003e in \u003cspan class=\\\"fbox\\\"\u003e\u003cimg src=\\\"https://images.houseofmath.com/overleaf-assets/images/gg-icon-menu-button.svg\\\" alt=\\\"GeoGebra icon\\\" width=\\\"19\\\" height=\\\"19\\\"\u003e\u003c/span\u003e \u003ccode\u003e\u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eMenu\u003c/span\u003e\u003c/code\u003e. Then use  \u003c/p\u003e \u003cdiv class=\\\"flushleft\\\"  \u003e \u003cp class=\\\"noindent\\\" \u003e   \u003cspan  class=\\\"ec-lmtt-12x-x-144\\\"\u003eAngle(\u0026#x003C;plane\u0026#x003E;, \u0026#x003C;plane\u0026#x003E;)\u003c/span\u003e \u003c/p\u003e\u003c/div\u003e \u003cp class=\\\"noindent\\\" \u003eto find the angle between the planes \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eα\u003c/mi\u003e\u003c/math\u003e and \u003cmath mathvariant=\\\"normal\\\"   xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"   display=\\\"inline\\\" \u003e\u003cmi  \u003eβ\u003c/mi\u003e\u003c/math type=\\\"punct\\\"\u003e. \u003c/p\u003e  \u003c/div\u003e  \u003c/div\u003e                                                                                                               \u003c/div\u003e \u003cp class=\\\"indent\\\" \u003e     \"},\"image\":null,\"isRootPath\":false,\"key\":\"content/en/software/02 GeoGebra/01 Numbers and Quantities/03 Vectors/02 Three Dimensions/04AnglesSpaceGG.tex\",\"language\":\"en\",\"locale\":\"en\",\"metaDescription\":\"Learn how to use GeoGebra to find the angle between two lines, a line and a plane or two planes, in 3D space. Discover the flexibility of this program.\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/find-angles-between-objects-in-space-with-geogebra\",\"previewUrl\":\"https://images.houseofmath.com/eyJrZXkiOiJvdmVybGVhZi1hc3NldHMvcHJldmlldy8wNGFuZ2xlc3NwYWNlZ2cucG5nIiwiYnVja2V0IjoiaG9tLWZyb250ZW5kLWFzc2V0cyIsInZlcnNpb24iOjN9\",\"sharedLanguageKey\":null,\"siblings\":[{\"h1\":\"How to Compute Vector Products with GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-compute-vector-products-with-geogebra\"},{\"h1\":\"How to Make Lines and Planes in GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-make-lines-and-planes-in-geogebra\"},{\"h1\":\"Find Angles Between Objects in Space with GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/find-angles-between-objects-in-space-with-geogebra\"},{\"h1\":\"How to Find Distance in GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-find-distance-in-geogebra\"},{\"h1\":\"How to Make a Sphere in GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-make-a-sphere-in-geogebra\"},{\"h1\":\"How to Find Intersection in GeoGebra\",\"path\":\"/geogebra/numbers-and-quantities/vectors/three-dimensions/how-to-find-intersection-in-geogebra\"}],\"taskReference\":null,\"title\":\"Find Angles Between Objects in Space with GeoGebra\",\"type\":\"geogebra\"}},\"bootcampVideo\":{\"progressPercentage\":0,\"videoLength\":1,\"isPlaying\":false,\"hasEnded\":false,\"isReset\":false},\"vcc\":{\"moduleCount\":1,\"title\":\"GeoGebra\",\"subjectName\":\"Geogebra\",\"complexityLevel\":1,\"whyShortTitle\":\"\",\"hasWhyLevel\":false,\"totalModules\":1,\"totalLevels\":0,\"levelsPageData\":[{\"hasEncyclopedia\":true,\"level\":\"1\",\"regular\":false,\"shortTitle\":\"What Are GeoGebra Views?\"},{\"hasEncyclopedia\":true,\"level\":\"2\",\"regular\":false,\"shortTitle\":\"How to Do Basic Math in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"3\",\"regular\":false,\"shortTitle\":\"The Keyboard Shortcuts and Settings of GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"4\",\"regular\":false,\"shortTitle\":\"How to Use the CAS Calculator in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"5\",\"regular\":false,\"shortTitle\":\"How to Use the Spreadsheet View in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"6\",\"regular\":false,\"shortTitle\":\"How to Insert GeoGebra content in an Exam Answer File\"},{\"hasEncyclopedia\":true,\"level\":\"7\",\"regular\":false,\"shortTitle\":\"How to Do Sequences in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"8\",\"regular\":false,\"shortTitle\":\"Use the Sum Command in GeoGebra for Sums and Series\"},{\"hasEncyclopedia\":true,\"level\":\"9\",\"regular\":false,\"shortTitle\":\"How to Find Prime Numbers with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"10\",\"regular\":false,\"shortTitle\":\"Find the Scientific Notation of a Number Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"11\",\"regular\":false,\"shortTitle\":\"How to Calculate Percent and Growth Factors in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"12\",\"regular\":false,\"shortTitle\":\"How to Use Prefixes in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"13\",\"regular\":false,\"shortTitle\":\"GeoGebra Template for Calculating Your Monthly Paycheck\"},{\"hasEncyclopedia\":true,\"level\":\"14\",\"regular\":false,\"shortTitle\":\"Spreadsheets for Budget and Accounting in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"15\",\"regular\":false,\"shortTitle\":\"Revenue, Cost, Unit Cost and Profit in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"16\",\"regular\":false,\"shortTitle\":\"Revenue with Demand and Price in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"17\",\"regular\":false,\"shortTitle\":\"How to Calculate Marginal Revenue with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"18\",\"regular\":false,\"shortTitle\":\"How to Find Cost-minimizing Input with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"19\",\"regular\":false,\"shortTitle\":\"How to Find Profit-maximizing Input with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"20\",\"regular\":false,\"shortTitle\":\"How to Graph a Vector in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"21\",\"regular\":false,\"shortTitle\":\"How to Graph a Position Vector in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"22\",\"regular\":false,\"shortTitle\":\"How to Graph Vector Sums and Differences in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"23\",\"regular\":false,\"shortTitle\":\"How to Check If Two Vectors are Parallel Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"24\",\"regular\":false,\"shortTitle\":\"How to Graph a Vector of Given Length in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"25\",\"regular\":false,\"shortTitle\":\"How to Compute Scalar Products of Vectors with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"26\",\"regular\":false,\"shortTitle\":\"Parametrization of Curves with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"27\",\"regular\":false,\"shortTitle\":\"How to Solve Vector Equations with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"28\",\"regular\":false,\"shortTitle\":\"Position, Velocity, Speed and Acceleration in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"29\",\"regular\":false,\"shortTitle\":\"How to Compute Vector Products with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"30\",\"regular\":false,\"shortTitle\":\"How to Make Lines and Planes in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"31\",\"regular\":false,\"shortTitle\":\"Find Angles Between Objects in Space with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"32\",\"regular\":false,\"shortTitle\":\"How to Find Distance in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"33\",\"regular\":false,\"shortTitle\":\"How to Make a Sphere in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"34\",\"regular\":false,\"shortTitle\":\"How to Find Intersection in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"35\",\"regular\":false,\"shortTitle\":\"Completing the Square Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"36\",\"regular\":false,\"shortTitle\":\"How to Solve an Equation in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"37\",\"regular\":false,\"shortTitle\":\"How to Solve Systems of Equations in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"38\",\"regular\":false,\"shortTitle\":\"Solve Multivariate Systems of Equations in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"39\",\"regular\":false,\"shortTitle\":\"How to Solve Inequalities in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"40\",\"regular\":false,\"shortTitle\":\"How to Compute Logarithms in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"41\",\"regular\":false,\"shortTitle\":\"Figures, Perimeter (Circumference) and Area in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"42\",\"regular\":false,\"shortTitle\":\"How to Draw Loci with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"43\",\"regular\":false,\"shortTitle\":\"Check If Two Figures are Congruent Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"44\",\"regular\":false,\"shortTitle\":\"How to Find Circumcircle and Circumcenter with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"45\",\"regular\":false,\"shortTitle\":\"How to Find Incircle and Incenter Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"46\",\"regular\":false,\"shortTitle\":\"How to Find Centroid and Medians with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"47\",\"regular\":false,\"shortTitle\":\"How to Find Orthocenter and Altitudes with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"48\",\"regular\":false,\"shortTitle\":\"How to Find the Euler Line of a Triangle in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"49\",\"regular\":false,\"shortTitle\":\"How to Make a Circle in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"50\",\"regular\":false,\"shortTitle\":\"How to Graph Circles of Apollonius in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"51\",\"regular\":false,\"shortTitle\":\"How to Study the Power of a Point Theorem with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"52\",\"regular\":false,\"shortTitle\":\"How to Graph an Ellipse in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"53\",\"regular\":false,\"shortTitle\":\"Translation of Figures in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"54\",\"regular\":false,\"shortTitle\":\"Constructing Parallels in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"55\",\"regular\":false,\"shortTitle\":\"Constructing a Parallel through a Point in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"56\",\"regular\":false,\"shortTitle\":\"How to Do a Reflection in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"57\",\"regular\":false,\"shortTitle\":\"Constructing the Tangents to a Circle in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"58\",\"regular\":false,\"shortTitle\":\"How to Create an Angle in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"59\",\"regular\":false,\"shortTitle\":\"How to Construct Supplementary Angles in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"60\",\"regular\":false,\"shortTitle\":\"How to Check if Two Angles are Equal with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"61\",\"regular\":false,\"shortTitle\":\"Inscribed and Central Angles in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"62\",\"regular\":false,\"shortTitle\":\"How to Rotate a Figure in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"63\",\"regular\":false,\"shortTitle\":\"Constructing the Normal Through a Point in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"64\",\"regular\":false,\"shortTitle\":\"How to Construct a Perpendicular Bisector in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"65\",\"regular\":false,\"shortTitle\":\"How to Construct an Angle Bisector in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"66\",\"regular\":false,\"shortTitle\":\"How to Make a Polyhedron in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"67\",\"regular\":false,\"shortTitle\":\"How to Draw Prisms in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"68\",\"regular\":false,\"shortTitle\":\"How to Draw Cylinders in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"69\",\"regular\":false,\"shortTitle\":\"How to Make Pyramids in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"70\",\"regular\":false,\"shortTitle\":\"How to Make a Cone in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"71\",\"regular\":false,\"shortTitle\":\"How to Make a Sphere in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"72\",\"regular\":false,\"shortTitle\":\"Surface Area and Volume of Solid Figures in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"73\",\"regular\":false,\"shortTitle\":\"How to Make a Unit Circle in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"74\",\"regular\":false,\"shortTitle\":\"How to Use Trigonometric Functions in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"75\",\"regular\":false,\"shortTitle\":\"Calculate with the Law of Sines and Cosines in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"76\",\"regular\":false,\"shortTitle\":\"How to Solve Trigonometric Equations in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"77\",\"regular\":false,\"shortTitle\":\"How to Graph a Trigonometric Function in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"78\",\"regular\":false,\"shortTitle\":\"Transforming into a Harmonic Oscillator in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"79\",\"regular\":false,\"shortTitle\":\"Perspective Drawing in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"80\",\"regular\":false,\"shortTitle\":\"How to Make a Frequency Table with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"81\",\"regular\":false,\"shortTitle\":\"How to Make a Bar Chart with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"82\",\"regular\":false,\"shortTitle\":\"How to Make a Pie Chart with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"83\",\"regular\":false,\"shortTitle\":\"How to Make a Line Chart with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"84\",\"regular\":false,\"shortTitle\":\"How to Make a Box Plot with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"85\",\"regular\":false,\"shortTitle\":\"How to Make a Histogram with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"86\",\"regular\":false,\"shortTitle\":\"How to Calculate Central Tendencies with GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"87\",\"regular\":false,\"shortTitle\":\"Relative Frequency and Law of Large Numbers in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"88\",\"regular\":false,\"shortTitle\":\"Compute Permutations and Combinations in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"89\",\"regular\":false,\"shortTitle\":\"How to Use GeoGebra as Binomial Distribution Calculator\"},{\"hasEncyclopedia\":true,\"level\":\"90\",\"regular\":false,\"shortTitle\":\"Hypergeometric Distribution Calculator in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"91\",\"regular\":false,\"shortTitle\":\"How to Use GeoGebra as a Normal Distribution Calculator\"},{\"hasEncyclopedia\":true,\"level\":\"92\",\"regular\":false,\"shortTitle\":\"Hypothesis Testing of Binomial Distribution in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"93\",\"regular\":false,\"shortTitle\":\"Hypothesis Testing with Normal Distribution in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"94\",\"regular\":false,\"shortTitle\":\"How to Graph a Function in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"95\",\"regular\":false,\"shortTitle\":\"How to Find Domain and Range of a Function in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"96\",\"regular\":false,\"shortTitle\":\"How to Find Intersection with the Axes in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"97\",\"regular\":false,\"shortTitle\":\"How to Find Intersection Between Graphs in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"98\",\"regular\":false,\"shortTitle\":\"How to Find the Zeros of a Function in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"99\",\"regular\":false,\"shortTitle\":\"How to Show Asymptotes in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"100\",\"regular\":false,\"shortTitle\":\"How to Draw a Tangent to a Curve in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"101\",\"regular\":false,\"shortTitle\":\"How to Animate Instantaneous Rate of Change in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"102\",\"regular\":false,\"shortTitle\":\"How to Calculate Limits of Functions Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"103\",\"regular\":false,\"shortTitle\":\"How to Find the Derivative of a Function in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"104\",\"regular\":false,\"shortTitle\":\"How to Find Stationary Points in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"105\",\"regular\":false,\"shortTitle\":\"How to Find Inflection Points in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"106\",\"regular\":false,\"shortTitle\":\"How to Use Integral in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"107\",\"regular\":false,\"shortTitle\":\"How to Find the Area Under a Graph Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"108\",\"regular\":false,\"shortTitle\":\"How to Find the Area Between Graphs Using GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"109\",\"regular\":false,\"shortTitle\":\"How to Find Solid of Revolution in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"110\",\"regular\":false,\"shortTitle\":\"Integral Curves and Direction Fields in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"111\",\"regular\":false,\"shortTitle\":\"How to Write a Differential Equation in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"112\",\"regular\":false,\"shortTitle\":\"How to Do Regression in GeoGebra\"},{\"hasEncyclopedia\":true,\"level\":\"113\",\"regular\":false,\"shortTitle\":\"How to Solve Linear Optimization Problems with GeoGebra\"}],\"levelActivities\":{\"1\":0,\"2\":0,\"3\":0,\"4\":0,\"5\":0,\"6\":0,\"7\":0,\"8\":0,\"9\":0,\"10\":0,\"11\":0,\"12\":0,\"13\":0,\"14\":0,\"15\":0,\"16\":0,\"17\":0,\"18\":0,\"19\":0,\"20\":0,\"21\":0,\"22\":0,\"23\":0,\"24\":0,\"25\":0,\"26\":0,\"27\":0,\"28\":0,\"29\":0,\"30\":0,\"31\":0,\"32\":0,\"33\":0,\"34\":0,\"35\":0,\"36\":0,\"37\":0,\"38\":0,\"39\":0,\"40\":0,\"41\":0,\"42\":0,\"43\":0,\"44\":0,\"45\":0,\"46\":0,\"47\":0,\"48\":0,\"49\":0,\"50\":0,\"51\":0,\"52\":0,\"53\":0,\"54\":0,\"55\":0,\"56\":0,\"57\":0,\"58\":0,\"59\":0,\"60\":0,\"61\":0,\"62\":0,\"63\":0,\"64\":0,\"65\":0,\"66\":0,\"67\":0,\"68\":0,\"69\":0,\"70\":0,\"71\":0,\"72\":0,\"73\":0,\"74\":0,\"75\":0,\"76\":0,\"77\":0,\"78\":0,\"79\":0,\"80\":0,\"81\":0,\"82\":0,\"83\":0,\"84\":0,\"85\":0,\"86\":0,\"87\":0,\"88\":0,\"89\":0,\"90\":0,\"91\":0,\"92\":0,\"93\":0,\"94\":0,\"95\":0,\"96\":0,\"97\":0,\"98\":0,\"99\":0,\"100\":0,\"101\":0,\"102\":0,\"103\":0,\"104\":0,\"105\":0,\"106\":0,\"107\":0,\"108\":0,\"109\":0,\"110\":0,\"111\":0,\"112\":0,\"113\":0},\"badgeImage\":{\"alt\":\"\",\"srcKey\":\"\"},\"badgeOutlineImage\":{\"alt\":\"\",\"srcKey\":\"\"},\"color\":{\"gradientClass\":\"bg-task-numbers-gradient\",\"bgClass\":\"bg-orchid-dark\",\"secondaryBgClass\":\"bg-orchid-dark\",\"checkedBgClass\":\"checked:bg-orchid-dark\",\"textClass\":\"text-orchid-dark\",\"hex\":\"#A840C8\",\"borderClass\":\"border-orchid-dark\",\"outlineClass\":\"outline-numbers\",\"ringClass\":\"ring-orchid-dark\",\"shadowBorderClass\":\"border-violet-darker\",\"shadowClass\":\"shadow-task-purple\"}},\"bootcampQuiz\":{\"bootcampKey\":null,\"subtopic\":null,\"startTime\":null,\"totalQuestions\":0,\"completedQuestions\":0,\"wrongAnswer\":0,\"bossHealth\":100,\"damagePlayer\":false,\"maxWrongAnswer\":0,\"questions\":[],\"mathEnergyEarned\":0,\"badgeImage\":{\"alt\":\"Special Trophy\",\"srcKey\":\"bootcamp/bootcamp_module_list_trophy_logo.png\"},\"bossBattleEnemy\":{\"avatarGLBFilePath\":\"\",\"avatarImagePath\":{\"alt\":\"\",\"srcKey\":\"\"},\"dragonId\":0},\"hasStartedQuiz\":false,\"bossCurrentBehavior\":null,\"bossIsBusy\":false,\"bossDamageAmount\":0,\"bossResponse\":{\"state\":\"ready\",\"anim\":null},\"bossSound\":\"unmute\"},\"cart\":{\"tutorForOther\":[{\"levelsOfEducation\":[\"grade-11-to-13\"],\"product\":null,\"subjects\":[\"math\"],\"includeTravelExpense\":false,\"location\":\"online\",\"email\":\"\",\"emailVerify\":\"\",\"firstName\":\"\",\"lastName\":\"\",\"uuid\":\"b6be9423-da82-42bd-9905-d89bac2b1961\"}],\"tutorForSelf\":{\"levelsOfEducation\":[\"grade-11-to-13\"],\"product\":null,\"subjects\":[\"math\"],\"includeTravelExpense\":false,\"location\":\"online\"},\"videoChat\":{\"levelsOfEducation\":[\"grade-11-to-13\"],\"product\":null,\"subjects\":[\"math\"],\"dateAsTimestamp\":null,\"needHelpWith\":\"\",\"scheduleAt\":null},\"bookingForSelfFormErrors\":{}},\"chat\":{\"isChatOpen\":false,\"openThreadId\":null,\"prevThreadId\":null,\"threadPos\":null,\"newMessage\":\"\",\"transitionDidFinish\":true,\"showChatInfoBox\":true,\"chatNotification\":false,\"lastMessage\":\"\",\"lastMessageSentAt\":null},\"chatAi\":{\"isLoading\":false,\"stopMessage\":false,\"isShowingChatInitialModal\":false,\"isChatAiOpen\":false,\"isHintMessage\":false,\"chatAiConsent\":false,\"threadMessages\":[],\"currentMessage\":{},\"modalPosition\":{\"x\":150,\"y\":205,\"width\":930,\"height\":550,\"minWidth\":420,\"minHeight\":350,\"disableDragging\":false,\"disableResizing\":false,\"padding\":72,\"messageWidth\":730,\"windowWidth\":0,\"windowHeight\":0,\"isInitialized\":false},\"errorMessageType\":null,\"threadId\":\"99f996ff-7bbc-4503-9c05-d2429de45223\",\"imageModal\":{\"isImageModalOpen\":false,\"image\":\"\"}},\"checkout\":{},\"dashboard\":{\"isTabToggled\":true,\"scrollToQuest\":false,\"highlightedItem\":null,\"isNavigateToLearningPath\":false,\"chestAnimation\":false,\"tutorial\":{\"product\":\"\",\"tutorialStep\":0,\"shouldShowTutorial\":false,\"shouldShowLearningPathTutorial\":false},\"scrollToActivityLearningPath\":false,\"openChecklist\":false,\"selectedGrade\":\"\",\"subjectOrder\":\"\",\"achievementPagePreviousUrl\":\"/dashboard\",\"stepTwoExpanded\":[]},\"dropdownMenu\":{\"menuToClose\":null,\"menuToOpen\":null},\"employee\":{\"fields\":{\"employees\":[],\"categories\":[]},\"errors\":{\"employees\":\"\",\"categories\":\"\"},\"backendError\":\"\"},\"gamification\":{\"notification\":false,\"isNotificationListOpen\":false,\"gems\":0,\"rewardsCount\":{\"initial\":0,\"collected\":0,\"available\":0},\"nextBootcampPage\":{\"currentActivityType\":0},\"hasFetchedEvents\":false,\"hasLeveledUp\":false,\"hasLoggedOut\":false,\"fromRetakeQuiz\":false,\"showHomebaseClosingX\":true,\"showShopButton\":true,\"isQuizFinished\":false,\"hasShownLeaderboardModal\":false},\"gemShop\":{\"isShowingGemShop\":false,\"didPurchase\":false,\"isLocked\":false},\"homebase\":{\"homebaseIsOpen\":false,\"scrollToBase\":false,\"isBadgeReady\":false,\"showingBadgeCollection\":false},\"login\":{\"fields\":{\"email\":\"\",\"password\":\"\"},\"errors\":{\"email\":\"\",\"password\":\"\"},\"backendError\":\"\",\"loginScenario\":\"default\",\"simpleUserProfile\":{},\"view\":\"default\"},\"recaptcha\":{\"captcha\":{\"challenge\":\"\",\"answer\":\"\"},\"authType\":null},\"homebaseQuiz\":{\"quizState\":[],\"advancedSubjects\":{\"articles\":[],\"question\":\"\"},\"quizView\":\"INITIAL\",\"completedQuizes\":0},\"mathVault\":{\"currentScreen\":\"activity\",\"answerState\":\"pending\",\"levelsSummary\":{},\"currentQuestionSummary\":{\"livesLeft\":2,\"questionNo\":null,\"numberOfTries\":-1,\"numberOfFailures\":-1,\"firstPassOnTryNo\":-1,\"passed\":false},\"selectedAnswer\":null,\"currentLocation\":{\"room\":\"OUTSIDE_VAULT\",\"chest\":null},\"kudosEarned\":0,\"showLogInOverlay\":false,\"answerBeforeLogIn\":{\"metadata\":null,\"nextActivityPath\":null},\"server\":{\"activity\":{\"_id\":\"\",\"metadata\":{\"language\":\"en\",\"chest\":\"\",\"tab\":1,\"elevator\":\"\",\"room\":\"\",\"metaDescription\":\"\",\"keywords\":\"\",\"globalId\":\"\",\"activityNumber\":1,\"roomColor\":\"purple\",\"numberOfTabs\":6,\"numberOfActivities\":10,\"reference\":\"\",\"references\":[],\"title\":\"\",\"points\":0,\"tabsActivity\":[],\"columnLayout\":\"one-column\"},\"question\":[],\"alternatives\":[],\"shouldShuffle\":false}},\"loading\":false,\"streak\":-1,\"navigations\":{\"prevLevel\":1,\"prevActivity\":1,\"nextLevel\":1,\"nextActivity\":1},\"pickedEgg\":false,\"pickedTempEgg\":false},\"menu\":{\"menuIsOpen\":false,\"menuIsExpanded\":false,\"hideHorizontalMenu\":false,\"horizontalMenuHeight\":0,\"showAllLearningTools\":false,\"showAllGames\":false},\"multipleChoiceTest\":{\"activeQuestion\":{\"index\":0,\"didAnswer\":false,\"didAnswerCorrectly\":false,\"selectedOption\":null},\"executeGemstoneAnimation\":false,\"questionsState\":null,\"view\":\"quiz\"},\"onboarding\":{\"chosenAccountType\":null,\"details\":{\"nickname\":\"\",\"email\":\"\",\"birthYear\":null,\"country\":\"\",\"wantsNewsletter\":false,\"skill\":null},\"errors\":{\"nickname\":\"\",\"email\":\"\",\"birthYear\":\"\",\"country\":\"\",\"wantsNewsletter\":\"\",\"skill\":\"\"},\"progress\":0,\"showHomMethods\":false},\"petState\":{\"petState\":{\"eggState\":0,\"dragonPresent\":false,\"eggChosen\":null,\"tempEggChosen\":null,\"name\":\"Mathagon\",\"level\":0,\"dragonRank\":0,\"lifeLength\":0,\"stillHatching\":true,\"kudosOrbs\":0,\"totalMathEnergy\":0,\"totalAffection\":0,\"currentMathEnergyLevel\":0,\"currentAffectionLevel\":50,\"lastTimeStamp\":0,\"hatchingCooldown\":0,\"hatchingEnergyCooldown\":[0,0,0,0,0],\"tricksLearned\":[],\"currentTrickLearning\":null,\"currentTrickProgress\":0,\"trickLearningCooldown\":0}},\"quest\":{\"questState\":{\"questName\":\"\",\"questRewards\":{\"kudos\":0,\"gems\":0,\"xp\":0},\"showCompletedQuestModal\":false},\"isNewQuest\":false,\"clickedQuest\":\"\",\"resize\":false,\"skipingToBoss\":false},\"order\":{},\"scheduleSession\":{\"info\":{\"location\":\"\"},\"time\":{\"startHours\":0,\"startMinutes\":0,\"endHours\":0,\"endMinutes\":0}},\"search\":{\"currentQuery\":\"\",\"historyOfQueries\":[],\"isModalOpen\":false,\"hasSearchBar\":false},\"paywall\":{\"returnPath\":\"/dashboard\"},\"statistics\":{\"showNicknameOverlay\":false,\"learningStatsFilter\":{\"timeFilter\":\"all-time\",\"productCategoryFilter\":\"all\"},\"rewardsStatsFilter\":\"all-time\",\"productCardTimeFilter\":\"last-30-days\",\"productCardFilterOptions\":[]},\"storyCard\":{\"id\":null},\"adminSubscription\":{\"stripeProductId\":\"\",\"stripePriceId\":\"\",\"country\":\"\",\"trialPeriodDays\":0,\"isDefault\":false,\"paymentGateways\":[]},\"theory\":{\"displayMultipleChoiceTest\":false},\"ui\":{\"previewSite\":false},\"modals\":{\"showFeedbackModal\":false,\"showSubscriptionActivityModal\":false,\"hasPurchaseTutoringPackages\":false,\"showAndroidAppModal\":false,\"showBurnZoneModal\":false},\"donation\":{\"donationDetails\":{\"donationAmount\":0,\"donationCurrency\":\"usd\",\"donationType\":\"one-time\"},\"donationDetailsErrors\":{\"donationAmount\":\"\",\"donationCurrency\":\"\",\"donationType\":\"\"}},\"userFeedback\":{\"rating\":null,\"product\":null,\"showUserFeedbackModal\":false,\"comment\":\"\"},\"notificationApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"notificationApi\"}},\"homApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":5,\"keepUnusedDataFor\":1,\"reducerPath\":\"homApi\"}},\"soundControl\":{\"muted\":false,\"volume\":1},\"blogApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"blogApi\"}},\"tutorOnDemandApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":true,\"keepUnusedDataFor\":60,\"reducerPath\":\"tutorOnDemandApi\"}},\"staticPageApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"staticPageApi\"}},\"activityApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"activityApi\"}},\"bootcampApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"bootcampApi\"}},\"craneGameApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"craneGameApi\"}},\"rewardApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"rewardApi\"}},\"dragonApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"dragonApi\"}},\"gemShopApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"gemShopApi\"}},\"usersCountApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"usersCountApi\"}},\"mathVaultActivityApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"mathVaultActivityApi\"}},\"mediaApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"mediaApi\"}},\"adminApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":true,\"keepUnusedDataFor\":1,\"reducerPath\":\"adminApi\"}},\"multipleChoiceTestApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"multipleChoiceTestApi\"}},\"questApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"questApi\"}},\"studytipsApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"studytipsApi\"}},\"settings\":{\"errors\":{},\"fields\":{\"email\":\"\",\"nickname\":\"\",\"country\":\"\",\"curriculum\":\"\",\"birthYear\":\"\",\"wantsNewsletter\":false,\"userType\":\"\",\"skill\":\"\",\"yearsInSchool\":\"\",\"gradeLevelId\":\"\",\"currPassword\":\"\",\"newPassword\":\"\",\"newPasswordConfirm\":\"\",\"phone\":\"\"},\"serverError\":null,\"isServerErrorInNicknameTab\":null,\"tab\":null},\"activitiesApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"activitiesApi\"}},\"collectionApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"collectionApi\"}},\"curriculumUserStatApi\":{\"queries\":{},\"mutations\":{},\"provided\":{},\"subscriptions\":{},\"config\":{\"online\":true,\"focused\":true,\"middlewareRegistered\":true,\"refetchOnFocus\":false,\"refetchOnReconnect\":false,\"refetchOnMountOrArgChange\":false,\"keepUnusedDataFor\":60,\"reducerPath\":\"curriculumUserStatApi\"}},\"checklist\":{\"rewardFeature\":null,\"featureImage\":null},\"curriculum\":{\"gradeListByCountry\":[],\"gradeContent\":{\"gradeLevel\":0,\"gradeLabel\":\"\",\"title\":\"\",\"isJuniorMathGrade\":false,\"hasSubdivision\":false,\"gradeSubdivisions\":[]},\"curriculumSubjectContent\":{\"topicId\":\"\",\"subjectLabel\":\"\",\"subjectOrder\":0,\"subjectUrl\":\"\",\"isFirstVisit\":false,\"competencyGoals\":[]},\"lastActivity\":{\"curriculumLevelId\":\"\",\"quarterNo\":0,\"gradeLevel\":0},\"curriculumChangeLoading\":false,\"curriculumMappingQuizQuestion\":{\"questionId\":\"\",\"quizId\":\"\",\"slug\":\"\",\"title\":\"\",\"description\":[],\"isLastQuestion\":false,\"isQuizComplete\":false},\"curriculumMappingQuizCompleted\":false,\"menuCurriculumDropdownExpanded\":false,\"gradePageCurriculumDropdownExpanded\":false,\"curriculumChangeWarning\":false,\"lockRedirection\":false,\"prerequisiteChestClicked\":false,\"pathToSubjectCurriculum\":\"/curriculum/onboarding/learning\"},\"persistCurriculum\":{\"activeCompetencyList\":[],\"scrollItemId\":\"\"}}"},"__N_SSG":true},"page":"/bootcamp/[subtopic]/[key]/[moduleKey]/[level]/[action]","query":{"subtopic":"curriculum","key":"geogebra","moduleKey":"1","level":"31","action":"definition"},"buildId":"MKvZb934U9f7d-bRH9X4p","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[76969],"gsp":true,"locale":"en","locales":["en","no","uk"],"defaultLocale":"en","scriptLoader":[]} \end{array}