Vectors are fundamental concepts in mathematics, particularly in the study of physics and engineering. Understanding vectors as arrows with direction and magnitude is essential for IB MYP 4-5 students, as it lays the groundwork for more advanced topics in vectors and transformations. This article delves into the representation, notation, and application of vectors, providing a comprehensive guide tailored to the IB MYP curriculum.

Key Concepts

1. What is a Vector?

A vector is a mathematical entity characterized by both **magnitude** (size) and **direction**. Unlike scalars, which possess only magnitude, vectors are essential in describing quantities that require direction for complete specification, such as velocity, force, and displacement.

2. Graphical Representation of Vectors

Vectors are often represented graphically as arrows. The **length** of the arrow corresponds to the vector's magnitude, while the **arrowhead** indicates its direction. For example, a vector representing a 5-meter displacement to the east would be depicted as an arrow pointing eastward with a length proportional to 5 meters.

3. Vector Notation

Vectors are typically denoted by boldface letters (e.g., v) or letters with an arrow on top (e.g., v). In component form, vectors are expressed in terms of their horizontal and vertical components, such as v = v_x **i** + v_y **j**, where **i** and **j** are unit vectors along the x and y axes, respectively.

4. Calculating Magnitude of a Vector

The magnitude of a vector v = v_x **i** + v_y **j** is calculated using the Pythagorean theorem:

$$ \| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2} $$

For example, if v_x = 3 and v_y = 4, then:

$$ \| \mathbf{v} \| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

5. Direction of a Vector

The direction of a vector is determined by the angle it makes with the positive x-axis. This angle, θ, can be found using trigonometric ratios:

$$ \theta = \arctan\left(\frac{v_y}{v_x}\right) $$

For instance, if v_x = 1 and v_y = 1, then:

$$ \theta = \arctan\left(\frac{1}{1}\right) = 45^\circ $$

6. Vector Addition

Vectors can be added using the **head-to-tail method** or by adding their corresponding components. If A = A_x **i** + A_y **j** and B = B_x **i** + B_y **j**, then:

$$ \mathbf{A} + \mathbf{B} = (A_x + B_x)\mathbf{i} + (A_y + B_y)\mathbf{j} $$

For example, if A = 2**i** + 3**j** and B = 1**i** + 4**j**, then:

$$ \mathbf{A} + \mathbf{B} = (2 + 1)\mathbf{i} + (3 + 4)\mathbf{j} = 3\mathbf{i} + 7\mathbf{j} $$

7. Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). If A = A_x **i** + A_y **j** and k is a scalar, then:

$$ k\mathbf{A} = (kA_x)\mathbf{i} + (kA_y)\mathbf{j} $$

For instance, multiplying A = 2**i** + 3**j** by k = 3:

$$ 3\mathbf{A} = (3 \times 2)\mathbf{i} + (3 \times 3)\mathbf{j} = 6\mathbf{i} + 9\mathbf{j} $$

8. Dot Product

The dot product (or scalar product) of two vectors A and B is a scalar calculated as:

$$ \mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y $$

This product is useful in finding the angle between vectors and determining orthogonality. For example, if A = 1**i** + 3**j** and B = 4**i** + 2**j**, then:

$$ \mathbf{A} \cdot \mathbf{B} = (1 \times 4) + (3 \times 2) = 4 + 6 = 10 $$

9. Applications of Vectors

Vectors have wide-ranging applications in various fields:

Physics: Describing forces, velocities, and accelerations.
Engineering: Analyzing structural loads and motion.
Computer Graphics: Rendering images and modeling movements.
Navigation: Determining directions and distances.

10. Challenges in Understanding Vectors

Students often encounter difficulties with vectors, such as:

Distinguishing between scalar and vector quantities.
Visualizing vectors in different dimensions.
Performing vector operations like addition, subtraction, and scalar multiplication.
Applying vectors to solve real-world problems.

Overcoming these challenges requires consistent practice and a solid grasp of the underlying principles.

Comparison Table

Aspect	Scalar	Vector
Definition	Quantity with only magnitude	Quantity with both magnitude and direction
Examples	Temperature, mass, speed	Velocity, force, displacement
Representation	Numerical value	Arrow in space
Operations	Addition, subtraction	Addition, subtraction, dot product, cross product
Representation in Equations	Single variable	Component form (e.g., v = v_xi + v_yj)

Summary and Key Takeaways

Vectors are essential in representing quantities with both magnitude and direction.
Understanding vector notation and graphical representation aids in visualizing and solving problems.
Key operations include vector addition, scalar multiplication, and dot product.
Vectors have diverse applications across various scientific and engineering fields.
Consistent practice is crucial for mastering vector concepts and operations.

Examiner Tip

Tips

Remember the mnemonic "SAVED" to differentiate vectors and scalars: Size, Angle, Vector, Direction. Practice drawing vectors to better visualize their components and directions. When performing operations, always keep track of your signs and directions to avoid calculation errors. For exams, familiarize yourself with vector formulas and apply them to various problems to build confidence.

Did You Know

Vectors play a crucial role in computer animations, enabling smooth and realistic movements in video games and movies. Additionally, the concept of vectors extends to higher dimensions, allowing mathematicians and scientists to solve complex problems in fields like quantum physics and machine learning. Interestingly, the ancient Greeks used vector-like concepts to describe forces in their studies of mechanics.

Common Mistakes

One common mistake is confusing scalar and vector quantities. For example, treating speed (scalar) as velocity (vector) can lead to incorrect conclusions. Another frequent error is improper vector addition, such as adding magnitudes without considering directions. Lastly, students often misapply the dot product, forgetting that it's a scalar resulting from multiplying corresponding components.

FAQ

What is the difference between a scalar and a vector?

A scalar has only magnitude, such as temperature or mass, while a vector has both magnitude and direction, like velocity or force.

How do you calculate the magnitude of a vector?

The magnitude is calculated using the Pythagorean theorem for 2D vectors: $$\| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2}$$.

What is vector addition?

Vector addition combines two vectors by adding their corresponding components, either graphically using the head-to-tail method or algebraically by summing the components.

Can vectors be multiplied by scalars?

Yes, scalar multiplication involves multiplying each component of a vector by the scalar, effectively changing its magnitude without altering its direction unless the scalar is negative.

What is the dot product of two vectors?

The dot product is a scalar obtained by multiplying corresponding components of two vectors and summing the results: $$\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y$$.

How are vectors used in real-world applications?

Vectors are used in various fields such as physics for representing forces, engineering for structural analysis, computer graphics for modeling movements, and navigation for determining directions.

1. Graphs and Relations

1.1 Transformations of Graphs

1.1.1 Translating Graphs Vertically and Horizontally

1.1.2 Stretching and Compressing Graphs

1.1.3 Reflections Across Axes

1.1.4 Combining Multiple Transformations

1.1.5 Describing Graph Changes Algebraically

1.2 Interpreting Real-Life Graphs

1.2.1 Reading Distance-Time Graphs

1.2.2 Interpreting Speed-Time (Velocity-Time) Graphs

1.2.3 Identifying Sections of Constant and Changing Rates

1.2.4 Connecting Graphs to Real Situations

1.2.5 Estimating Values and Trends from Graphs

1.3 Cartesian Plane and Coordinates

1.3.1 Plotting Points in Four Quadrants

1.3.2 Identifying Coordinates from a Graph

1.3.3 Midpoint and Distance Between Two Points

1.3.4 Real-Life Uses of Cartesian Coordinates

1.3.5 Interpreting Grids and Map Coordinates

1.4 Graphing Linear Equations

1.4.1 Drawing Graphs from y = mx + c

1.4.2 Identifying Slope and Y-Intercept

1.4.3 Table of Values and Line Plotting

1.4.4 Interpreting and Writing Linear Equations from Graphs

1.4.5 Parallel and Perpendicular Lines (Introductory)

1.5 Graphing Quadratic, Cubic, and Exponential Functions

1.5.1 Sketching y = ax² + bx + c

1.5.2 Finding Vertex and Axis of Symmetry

1.5.3 Understanding Graph Shape and Turning Points

1.5.4 Graphing Simple Cubic Functions

1.5.5 Graphing and Identifying Exponential Growth and Decay

1.6 Parallel and Perpendicular Lines

1.6.1 Gradient of Parallel Lines

1.6.2 Gradient of Perpendicular Lines

1.6.3 Using Slope Relationships in Graphs

1.6.4 Finding Equations of Lines Given Conditions

1.6.5 Verifying Line Relationships Algebraically

2. Statistics and Probability

2.1 Experimental vs Theoretical Probability

2.1.1 Conducting Experiments and Simulations

2.1.2 Relative Frequency and Long-Term Trends

2.1.3 Comparing Experimental and Expected Results

2.1.4 Fairness, Bias, and Randomness

2.1.5 Applications in Real-World Risk Assessment

2.2 Collecting and Organizing Data

2.2.1 Types of Data: Discrete vs Continuous

2.2.2 Methods of Data Collection and Sampling

2.2.3 Frequency Tables and Two-Way Tables

2.2.4 Designing Surveys and Questionnaires

2.2.5 Organizing Data for Effective Analysis

2.3 Independent and Dependent Events

2.3.1 Defining Independence in Probability

2.3.2 Recognizing and Modeling Dependent Events

2.3.3 Conditional Probability (Introductory)

2.3.4 Using Probability Trees for Multi-Stage Events

2.3.5 Real-Life Applications: Games, Genetics, and Decision-Making

2.4 Measures of Central Tendency

2.4.1 Calculating Mean, Median, and Mode

2.4.2 Choosing the Appropriate Measure

2.4.3 Mean from a Frequency Table

2.4.4 Comparing Data Sets Using Averages

2.4.5 Real-Life Interpretation of Averages

2.5 Measures of Dispersion

2.5.1 Range and Interquartile Range

2.5.2 Standard Deviation (Introductory Concepts)

2.5.3 Box and Whisker Plots for Spread of Data

2.5.4 Comparing Variability Between Data Sets

2.5.5 Impact of Outliers on Dispersion Measures

2.6 Box Plots, Histograms, and Cumulative Frequency

2.6.1 Constructing and Interpreting Box Plots

2.6.2 Drawing and Reading Histograms

2.6.3 Using Class Intervals in Histograms

2.6.4 Cumulative Frequency Graphs and Curves

2.6.5 Finding the Median and Quartiles from Cumulative Graphs

2.7 Probability Rules and Tree Diagrams

2.7.1 Theoretical Probability and Sample Space Diagrams

2.7.2 Complementary Events and Basic Rules

2.7.3 Tree Diagrams for Independent Events

2.7.4 Tree Diagrams for Dependent Events

2.7.5 Calculating Combined Probabilities

3. Trigonometry

3.1 Solving Triangles with Given Information

3.1.1 Working with Two Sides and Included Angle (SAS)

3.1.2 Solving ASA and AAS Triangle Situations

3.1.3 Using Given Side and Opposite Angle (SSA)

3.1.4 Multiple Solutions in Ambiguous SSA Cases

3.1.5 Real-Life Applications in Engineering and Design

3.2 Applications in Bearings and Navigation

3.2.1 Understanding Bearings as Directional Angles

3.2.2 Solving Bearings Questions Using Trigonometry

3.2.3 Drawing Diagrams from Bearings Descriptions

3.2.4 Multi-Step Navigation Problems

3.2.5 Using Scale and Bearings Together

3.3 Trigonometric Graphs and Transformations

3.3.1 Sketching Graphs of y = sin x, cos x, tan x

3.3.2 Amplitude, Period, and Phase Shift

3.3.3 Transforming Trig Graphs Algebraically

3.3.4 Graphing Real-Life Periodic Phenomena

3.3.5 Solving Equations Graphically

3.4 Trigonometric Ratios in Right Triangles

3.4.1 Defining Sine, Cosine, and Tangent

3.4.2 Using SOH-CAH-TOA to Find Missing Sides

3.4.3 Using Trig Ratios to Find Missing Angles

3.4.4 Solving Word Problems with Right-Angled Triangles

3.4.5 Using Trigonometry with Bearings and Heights

3.5 Using Trigonometry in 3D Problems

3.5.1 Identifying Right Triangles in 3D Solids

3.5.2 Calculating Diagonals and Angles in Cuboids

3.5.3 Combining Pythagoras and Trig in 3D Geometry

3.5.4 Applications in Architecture and Physics

3.5.5 Word Problems Involving 3D Trigonometric Reasoning

3.6 Sine Rule, Cosine Rule, and Area of a Triangle

3.6.1 Conditions for Using the Sine Rule

3.6.2 Applying the Cosine Rule to Find Sides or Angles

3.6.3 Choosing Between Sine Rule and Cosine Rule

3.6.4 Finding the Area of Any Triangle Using Trigonometry

3.6.5 Problem Solving with Non-Right Triangles

4. Algebraic Expressions and Identities

4.1 Algebraic Fractions and Rational Expressions

4.1.1 Simplifying Algebraic Fractions

4.1.2 Multiplying and Dividing Algebraic Fractions

4.1.3 Adding and Subtracting with Common Denominators

4.1.4 Algebraic Fraction Word Problems

4.1.5 Restrictions and Undefined Values in Expressions

4.2 Laws of Exponents and Algebraic Manipulation

4.2.1 Product, Quotient, and Power Laws

4.2.2 Negative and Zero Exponents

4.2.3 Exponents in Algebraic Equations

4.2.4 Simplifying Complex Exponential Expressions

4.2.5 Word Problems Involving Exponents

4.3 Special Identities and Common Patterns

4.3.1 Perfect Square Identities

4.3.2 Difference of Squares Identity

4.3.3 Sum and Difference of Cubes (Introductory)

4.3.4 Recognizing and Using Identities in Problems

4.3.5 Deriving Identities from Expansion and Factorization

4.4 Algebraic Simplification and Expansion

4.4.1 Simplifying Like Terms

4.4.2 Expanding Single Brackets

4.4.3 Expanding Double Brackets (Binomials)

4.4.4 Simplifying Expressions with Multiple Terms

4.4.5 Combining Like Terms After Expansion

4.5 Substitution and Evaluation of Expressions

4.5.1 Substituting Numbers into Algebraic Expressions

4.5.2 Evaluating with Negative Numbers and Fractions

4.5.3 Solving Word Problems with Substitution

4.5.4 Using Substitution in Formulas

4.5.5 Multiple Substitutions in Complex Expressions

4.6 Factorization Techniques

4.6.1 Factoring Out the Greatest Common Factor

4.6.2 Factoring Trinomials (Quadratic Form)

4.6.3 Factoring Difference of Squares

4.6.4 Grouping and Advanced Patterns

4.6.5 Choosing Appropriate Factorization Methods

5. Geometry and Measurement

5.1 Construction and Loci

5.1.1 Constructing Perpendicular and Angle Bisectors

5.1.2 Constructing Triangles with Given Conditions

5.1.3 Using Compass and Ruler for Geometric Constructions

5.1.4 Loci from Fixed Distances or Rules

5.1.5 Shading Regions Using Multiple Loci

5.2 Bearings and Scale Drawings

5.2.1 Understanding Bearings in Navigation

5.2.2 Measuring and Drawing Bearings

5.2.3 Using Bearings with Trigonometry

5.2.4 Creating and Interpreting Scale Diagrams

5.2.5 Word Problems Involving Bearings and Maps

5.3 Lines, Angles, and Polygons

5.3.1 Types of Angles and Angle Relationships

5.3.2 Angles on a Line and Around a Point

5.3.3 Parallel Lines and Transversals

5.3.4 Interior and Exterior Angles of Polygons

5.3.5 Properties of Regular and Irregular Polygons

5.4 Units, Conversions, and Compound Measures

5.4.1 Converting Between Metric Units

5.4.2 Time, Speed, and Distance Problems

5.4.3 Density, Mass, and Volume

5.4.4 Pressure = Force ÷ Area Applications

5.4.5 Multi-Step Problems with Compound Measures

5.5 Congruence and Similarity

5.5.1 Criteria for Triangle Congruence (SSS, SAS, ASA, RHS)

5.5.2 Using Congruent Triangles in Proofs

5.5.3 Criteria for Similar Triangles

5.5.4 Using Scale Factor in Similar Figures

5.5.5 Real-Life Applications of Similarity and Congruence

5.6 Circles and Angle Properties

5.6.1 Naming Parts of a Circle

5.6.2 Angles at the Centre and in the Circle

5.6.3 Angles in the Same Segment and Alternate Segment Theorem

5.6.4 Cyclic Quadrilaterals and Chords

5.6.5 Tangent Properties and Circle Geometry Proofs

6. Equations, Inequalities, and Formulae

6.1 Linear Inequalities and Graphical Solutions

6.1.1 Solving Linear Inequalities in One Variable

6.1.2 Representing Solutions on a Number Line

6.1.3 Using Inequality Symbols Correctly

6.1.4 Graphical Representation of Inequalities

6.1.5 Inequality Word Problems

6.2 Simultaneous Linear Equations

6.2.1 Solving by Substitution Method

6.2.2 Solving by Elimination Method

6.2.3 Solving Graphically

6.2.4 Applications in Real-Life Problems

6.2.5 Checking Solutions for Both Equations

6.3 Quadratic Equations

6.3.1 Solving by Factorization

6.3.2 Solving Using the Quadratic Formula

6.3.3 Solving by Completing the Square (Introductory)

6.3.4 Nature of Roots Using Discriminant

6.3.5 Word Problems Involving Quadratic Equations

6.4 Literal Equations and Rearranging Formulae

6.4.1 Understanding Literal Equations

6.4.2 Rearranging to Make a Variable the Subject

6.4.3 Rearranging with Fractions and Brackets

6.4.4 Applications in Geometry and Physics Formulas

6.4.5 Common Mistakes in Rearranging

6.5 Real-Life Problem Solving with Equations

6.5.1 Translating Word Problems into Equations

6.5.2 Multi-Step Problems with Variables

6.5.3 Age, Distance, and Geometry-Based Problems

6.5.4 Choosing Appropriate Methods

6.5.5 Interpreting and Validating Solutions

6.6 Linear Equations in One Variable

6.6.1 Solving One-Step and Two-Step Equations

6.6.2 Equations with Variables on Both Sides

6.6.3 Equations Involving Brackets and Fractions

6.6.4 Word Problems Involving Linear Equations

6.6.5 Checking Solutions and Substitution

7. Number and Operations

7.1 Error Bounds and Approximations

7.1.1 Rounding to Decimal Places and Significant Figures

7.1.2 Upper and Lower Bounds

7.1.3 Calculations Using Rounded Values

7.1.4 Understanding Absolute and Percentage Error

7.1.5 Estimation in Problem Solving

7.2 Real Number System and Classifications

7.2.1 Classifying Rational and Irrational Numbers

7.2.2 Understanding Integers, Fractions, and Decimals

7.2.3 Comparing and Ordering Real Numbers

7.2.4 Set Notation and Venn Diagrams for Number Sets

7.2.5 Number Line Representation of Real Numbers

7.3 Financial Mathematics

7.3.1 Simple and Compound Interest

7.3.2 Profit, Loss, and Discount Calculations

7.3.3 Currency Conversion and Exchange Rates

7.3.4 Taxation, Budgeting, and Loan Payments

7.3.5 Solving Real-World Finance Problems

7.4 Operations with Fractions and Decimals

7.4.1 Addition and Subtraction of Fractions

7.4.2 Multiplication and Division of Fractions

7.4.3 Decimal Operations with Accuracy

7.4.4 Converting Between Fractions, Decimals, and Percentages

7.4.5 Order of Operations with Mixed Numbers and Decimals

7.5 Estimation, Rounding, and Significant Figures

7.5.1 Estimation Techniques in Mental Math

7.5.2 Rounding Large and Small Numbers

7.5.3 Significant Figures in Measurement

7.5.4 Use of Estimation in Checking Results

7.5.5 Approximations in Scientific and Everyday Contexts

7.6 Exponents, Powers, and Roots

7.6.1 Laws of Exponents for Integer Powers

7.6.2 Negative and Zero Exponents

7.6.3 Square Roots and Cube Roots

7.6.4 Simplifying Expressions with Exponents

7.6.5 Rational Exponents and Surds (Introductory)

7.7 Scientific Notation and Standard Form

7.7.1 Writing Large and Small Numbers in Scientific Notation

7.7.2 Converting Between Forms

7.7.3 Multiplying and Dividing Numbers in Scientific Notation

7.7.4 Adding and Subtracting with Powers of 10

7.7.5 Applications in Science and Engineering Contexts

7.8 Ratio, Proportion, and Percentages

7.8.1 Simplifying Ratios and Comparing Quantities

7.8.2 Solving Direct and Inverse Proportion Problems

7.8.3 Percentage Increase and Decrease

7.8.4 Compound Percentages and Successive Changes

7.8.5 Solving Word Problems Involving Percentages

8. Sequences, Patterns, and Functions

8.1 Number Patterns and Arithmetic Sequences

8.1.1 Recognizing and Continuing Arithmetic Sequences

8.1.2 Finding the Common Difference

8.1.3 Deriving the nth Term of an Arithmetic Sequence

8.1.4 Solving Problems Using Arithmetic Sequences

8.1.5 Real-World Applications of Arithmetic Patterns

8.2 Composite and Inverse Functions (Introductory)

8.2.1 Understanding Composite Functions Notation

8.2.2 Evaluating Composite Functions

8.2.3 Concept of Inverse Functions

8.2.4 Finding Inverses Algebraically (Intro)

8.2.5 Checking if Two Functions Are Inverses

8.3 Geometric Sequences and Series

8.3.1 Recognizing Geometric Sequences

8.3.2 Finding the Common Ratio

8.3.3 General Term of a Geometric Sequence

8.3.4 Solving Word Problems with Geometric Sequences

8.3.5 Sum of a Finite Geometric Series (Introductory)

8.4 General Term (nth Term) of a Sequence

8.4.1 Distinguishing Between Linear and Non-Linear Sequences

8.4.2 Creating a Rule for the nth Term

8.4.3 Evaluating Terms Using the nth Term Formula

8.4.4 Finding Term Position Given Value

8.4.5 Applications in Tables and Graphs

8.5 Function Notation and Interpretation

8.5.1 Understanding Function Notation f(x)

8.5.2 Evaluating Functions for Given Inputs

8.5.3 Interpreting Function Values in Context

8.5.4 Defining Piecewise and Step Functions (Introductory)

8.5.5 Using Function Machines and Diagrams

8.6 Domain and Range of Functions

8.6.1 Identifying Domain and Range from Tables

8.6.2 Determining Domain and Range from Graphs

8.6.3 Writing Inequality Notation for Domain and Range

8.6.4 Identifying Discrete and Continuous Functions

8.6.5 Applications in Real-World Graphs

9. Mensuration

9.1 Volume and Surface Area of Pyramids, Cones, and Spheres

9.1.1 Surface Area of Cones and Spheres

9.1.2 Volume of Cones, Spheres, and Hemispheres

9.1.3 Surface Area and Volume of Pyramids

9.1.4 Comparing Volumes of Different Solids

9.1.5 Solving Composite Solid Problems

9.2 Density, Mass, and Volume Problems

9.2.1 Understanding the Density Formula

9.2.2 Calculating Unknowns in the Density Formula

9.2.3 Changing Units in Density Problems

9.2.4 Applying Density to Real-World Situations

9.2.5 Multi-Step Problems with Density and Volume

9.3 Trapeziums, Sectors, and Composite Figures

9.3.1 Area of Circular Sectors and Segments

9.3.2 Arc Length and Sector Area Calculations

9.3.3 Combining Shapes: Semi-Circles, Rectangles, Trapeziums

9.3.4 Estimating and Approximating Irregular Areas

9.3.5 Challenging Geometry Problems Involving Composite Figures

9.4 Perimeter and Area of 2D Shapes

9.4.1 Perimeter of Triangles, Quadrilaterals, and Circles

9.4.2 Area of Rectangles, Triangles, and Parallelograms

9.4.3 Area of Trapeziums and Rhombuses

9.4.4 Using Compound Shapes in Area Calculations

9.4.5 Solving Real-World Problems Involving Area and Perimeter

9.5 Surface Area and Volume of 3D Solids

9.5.1 Nets and Surface Area of Cubes and Cuboids

9.5.2 Surface Area of Cylinders and Prisms

9.5.3 Volume of Cubes, Cuboids, and Cylinders

9.5.4 Real-Life Applications of Volume and Surface Area

9.5.5 Units and Conversion in Volume Calculations

10. Vectors and Transformations

10.1 Vector Arithmetic and Scalar Multiplication

10.1.1 Adding and Subtracting Vectors Graphically

10.1.2 Adding and Subtracting Column Vectors

10.1.3 Scalar Multiplication of Vectors

10.1.4 Vector Arithmetic Word Problems

10.1.5 Simplifying and Combining Vectors

10.2 Position Vectors and Displacement

10.2.1 Definition and Use of Position Vectors

10.2.2 Finding Displacement Between Two Points

10.2.3 Using Coordinates to Calculate Displacement

10.2.4 Position Vectors in Geometric Diagrams

10.2.5 Describing Movement in Terms of Vectors

10.3 Geometrical Transformations

10.3.1 Describing and Performing Translations

10.3.2 Reflections in Lines: x-axis, y-axis, y = x, y = -x

10.3.3 Rotations About a Point and Angle

10.3.4 Enlargement with Positive and Negative Scale Factors

10.3.5 Combining Multiple Transformations

10.4 Combined Transformations and Matrices

10.4.1 Using Matrices for Simple Transformations (Introductory)

10.4.2 Applying Successive Transformations

10.4.3 Identifying Single Equivalent Transformation

10.4.4 Matrix Representation of Translations

10.4.5 Transforming Coordinates with Matrix Multiplication

10.5 Vector Representation and Notation

10.5.1 Understanding Vectors as Arrows with Direction and Magnitude

10.5.2 Column Vector Notation

10.5.3 Representing Vectors in Diagrams and Coordinates

10.5.4 Notation and Labeling for Displacement Vectors

10.5.5 Unit Vectors and Position Vectors (Introductory)

10.6 Vector Proofs and Applications

10.6.1 Proving Collinearity Using Vectors

10.6.2 Midpoints and Parallel Vectors

10.6.3 Geometric Proofs with Vectors

10.6.4 Solving Complex Vector Path Problems

10.6.5 Applications in Navigation and Design

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Understanding Vectors as Arrows with Direction and Magnitude

Introduction