Numbers and QuantitiesMenu1The Decimal System and the Numbers up to 1002What Is the Real Number Line?3How to Compare Numbers by Size4Pairs of Ten and Groups of Ten up to 100Activity Set 15What Are Even and Odd Numbers?6What Is the Place-value System?Activity Set 17Sets of Numbers8Interval and Set Notation9What Are Number Sequences?10Examples of Mathematical Sequences11How Do You Find the n-th Term in a Sequence?12How Do Arithmetic Sequences Work?13Square Numbers, Cube Numbers and Triangular Numbers14Recursive and Explicit Formulas for Sequences15What Are Series in Math?16What Are Arithmetic Series?17How to Calculate Amortizing Loans with Series Formulas18Geometric Series Formulas19Infinite Geometric Series and Convergence20How Is Fixed-rate Mortgage Calculated?21How to Calculate Present Values Using Timelines22How to Calculate Future Values Using Timelines23Examples of Calculations with Series24What Is Decimal Notation?25Which Number Is in the Tenths Place?26Which Number Is in the Hundredths Place?27What Does Negative Numbers Mean?28How to Calculate with Negative Numbers29Prime NumbersActivity Set 130Prime Factorization Using the Division Method31Prime Factorization Using a Factor Tree32How to Write Scientific Notation33How to Calculate with Scientific Notation34How to Convert Scientific Notation to Standard Notation35What Are Complex Numbers?36What Does the Complex Plane Mean?37What Is Polar Form for Complex Numbers?38What Is the Norm and the Argument of a Complex Number?39What Is Euler’s Formula for Complex Numbers?40What Are Complex Conjugates?41Addition and Subtraction of Complex NumbersActivity Set 142How to Multiply Complex Numbers43How Can You Simplify Complex Fractions44What Are the Roots of a Complex Number?45What Is de Moivre's Formula and How Do You Use It?46How to Find the Distance Between Two Complex Numbers47How to Find the Midpoint of Two Complex Numbers48How to Solve Linear Equations with Complex Numbers49How to Solve Quadratic Equations with Complex Numbers50How to Factorize Complex Polynomials51What Is the Fundamental Theorem of Algebra?52Addition with Help from the Number Line53Subtraction with Help from the Number Line54Addition and Subtraction (The Place-value System)55How to Add Using the Column Method56How to Subtract Using the Column Method57Multiplication as Repeated Addition58The Times Tables59Which Products Give the Answers in the Times Tables?60How to Multiply by 1061How to Multiply by 10062How to Multiply by 100063How Long Multiplication Works64What Is Doubling in Math?65What Does Halving Mean in Math?66Division as Inverse Multiplication67What Are the Divisibility Rules?68How to Divide by 1069How to Divide by 10070How to Divide by 100071How to Do Long Division72Examples of Long Division73What Are Powers in Math?74What Are the Power Rules?75What Are Powers of 10?76How Square Roots Work77Square Roots of Products and Fractions78Order of Operations (MDAS)79Parentheses and Order of Operations80Order of Operations (PEMDAS)81Signs in Math (Positive or Negative)82How to Do Mental Math (Rounding)83How to Do Mental Math (Estimates)84How to Round a Number85How to Make Estimates (Mental Math)86How to Calculate the Cost per Unit87What Are Fractions?88What Is the Value of a Fraction?89Proper Fraction, Improper Fraction and Mixed Fraction90How Do You Expand Fractions?91How Do You Simplify Fractions?92How to Simplify a Fraction by Canceling93Add and Subtract Fractions with Equal Denominators94Add and Subtract Fractions with Different Denominators95Find the Common Denominator by Multiplying Denominators96Find the Common Denominator by Expanding or Simplifying97Find the Common Denominator Using the Times Tables98How to Find the Common Denominator Using Factorization99How to Multiply Fractions100How to Multiply Fractions by Cross-canceling101How to Divide Fractions102What Does Percentage Mean?103How Can You Calculate Percentages?104How to Find Relative Percent Change105How to Calculate Growth Factor106How to Use Growth Factors in Calculations107How Do Percentage Points Work?108Convert Percentages to Fractions and Decimal Numbers109Convert Fractions and Decimal Numbers to Percentages110How to Convert Between Decimal Numbers and Fractions111Converting Fractions, Decimal Numbers and Percentages112What Are the Units of Measurement in the Metric System?113What Are the Metric Units for Length?114What Are the Metric Units for Area?115What Are the Metric Units for Volume?116What Are the Metric Units for Weight?117What Are the Prefixes for SI Units?118How to Find Volumetric Mass Density (Metric)119What Is Temperature and What Are Degrees Celsius?120How Do You Calculate Metric Units of Measurement?121Choosing Appropriate Metric Units of Measurement122What Are the Units of Measurement in the Imperial System?Activity Set 1123What Are the Imperial Units for Length?124What Are the Imperial Units for Area?Activity Set 1125What Are the Imperial Units for Volume?Activity Set 1126What Are the Imperial Units for Weight?Activity Set 1127How to Find Volumetric Mass Density (Imperial)128What Is Temperature and What Are Degrees Fahrenheit?129How Do You Calculate Imperial Units of Measurement?130How Do You Convert Imperial to Metric?131How Do You Convert Metric to Imperial?132The Digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)133How Do Significant Figures Work?134What Does Measurement Uncertainty Mean?135What Does Relative Measurement Uncertainty Mean?136What Is Digital and Analog Clock?137Learning About the Clock (Seconds)138Learning About the Clock (Minutes)Activity Set 1139Learning About the Clock (Hours)Activity Set 1140Learning About the Clock (Whole and Half Hours)141Learning About the Clock (Quarters of an Hour)142Learning About the Clock (Ten To and Ten Past)143Learning About the Clock (Five To and Five Past)144Mental Arithmetic with Time145Calculations with the Clock146What Is Day and Night?147What Are Days and Weeks?148What Are Months?149What Are the Seasons of the Year?150How Does Scale Ratio Work?151How to Read Blueprint Scale152What Is the Ratio Between Two Numbers?153What Does Mixing Ratio Mean?154How to Find Distance if You Have Velocity and Time155What Is a Speed Diagram?156Coins of the United States Dollar157Banknotes of the United States Dollar158How Do You Count Money?159How Math Is Used in Shopping160How Currency Exchange Works161How Can Personal Finance Best Be Managed?162What Are the Different Types of Income?163How Does Commission Salary Work?164How Does Tax and Tax Basis Work?165How Do Bonus Payments Work?166How Value-added Tax Is Calculated167How Credit Card Interest Rates Work168Consumer Price Index and Purchasing Power of a Dollar169What Are Price Index and Consumer Price Index?170What Are Real Wages in Economics?171How Do Loans Work?172What Is an Amortizing Loan?173What Are Fixed-rate Mortgages?174How to Budget Your Money175How to Use Spreadsheets for Accounting176Interest Over One Period177Interest Over Several Periods178How to Compute Compound Interest on Savings179What Do Revenue, Cost, Unit Cost and Profit Mean?180How Do Price and Demand Affect Revenue?181Marginal Revenue, Marginal Cost and Marginal Profit182How to Find Cost-minimizing Input183How to Find Profit-maximizing Input184What Is a Vector?185What Does Vector Coordinates Mean?186What is a Position Vector?Activity Set 1187Sum and Difference of VectorsActivity Set 1188Numbers multiplied by a vectorActivity Set 1189How to Find the Vector Between Two Points190What Is the Length of a Vector?191Finding the Distance Between Two Points Using Vectors192How to Find Parallel Vectors193How to Find the Scalar Product of Two Vectors (2D)194How Can You Find the Angle Between Two Vectors? The angle between two vectors is always an angle α ∈[0∘, 180∘]. Formula The Angle Between Two Vectors cos α = u→ ⋅v→ |u→| ⋅|v→|,α ∈[0∘, 180∘] Example 1 Find the angle between the vectors (4, 1) and (2,−3). You insert the numbers into the formula, which gives you cos α = (4, 1) ⋅(2,−3) | (4, 1)| ⋅|(2,−3)| = 8 − 3 42 + 12 ⋅22 + (−3 ) 2 = 5 17 ⋅13 ≈ 5 14.86 ≈ 0.336. That means the angle is α = cos −1 (0.336) ≈ 70.3∘. Want to know more?Sign UpIt's free!DoneContinue195What Does Unit Vector Mean?196What Are Basis Vectors Used For?197Find the Medians of a Triangle Using Vector Calculation198How to Parametrize a Line199How to Solve Vector Equations200What Are Vector Functions?201Vectors in Three-dimensional Spaces202How to Find the Scalar Product of Two Vectors (3D)203What Is the Vector Product?204How Does the Right-hand Rule for Cross Products Work?205How to Calculate Area of a Parallelogram with Vectors206How to Calculate Area of a Triangle with Vectors207Calculate Volume of Prism and Pyramid with Vectors208What Does Parameterization Mean?209Parameterization of Lines in Space210Parameterization of a Plane211How to Determine the Angle Between Two Lines212How Do You Find the Angle Between a Line and a Plane?213How to Find the Angle Between Two Planes214How to Find the Distance from a Point to a Line215How to Find the Distance Between Two Lines216How to Find the Distance Between a Point and a Plane217How to Find the Distance Between a Line and a Plane218How to Find the Distance Between Two Planes219What Is the Equation for a Sphere?220How to Find the Surface Area of a Spherical Segment221The Intersection Between Planes and Coordinate Axes222How to Find the Intersection Between Line and Plane223How to Find the Intersection Between Two Planes224Intersection Between a Tangent Plane and a Sphere225The Intersection Between a Plane and a Sphere
The angle between two vectors is always an angle α ∈[0∘, 180∘]. Formula The Angle Between Two Vectors cos α = u→ ⋅v→ |u→| ⋅|v→|,α ∈[0∘, 180∘] Example 1 Find the angle between the vectors (4, 1) and (2,−3). You insert the numbers into the formula, which gives you cos α = (4, 1) ⋅(2,−3) | (4, 1)| ⋅|(2,−3)| = 8 − 3 42 + 12 ⋅22 + (−3 ) 2 = 5 17 ⋅13 ≈ 5 14.86 ≈ 0.336. That means the angle is α = cos −1 (0.336) ≈ 70.3∘. Want to know more?Sign UpIt's free!DoneContinue
