Chapter 8: Vectors

Overview

Vectors are mathematical objects that possess both magnitude (size) and direction, distinguishing them from scalars which have only magnitude. This chapter explores fundamental vector concepts including vector notation, vector operations, and vector representation in the Cartesian plane. Mastery of vectors is essential for understanding physics, engineering, and advanced mathematics topics like linear algebra and calculus.

Learning Objectives

After completing this chapter, you will be able to:

• Understand and use vector notation
• Perform vector addition and subtraction
• Calculate vector magnitudes
• Represent vectors in Cartesian form
• Find unit vectors in given directions
• Apply vector concepts to solve geometric problems

Key Concepts

8.1 Vectors

Definition of Vectors

A vector is a quantity that has both magnitude (length) and direction. Examples include displacement, velocity, force, and acceleration.

A scalar is a quantity that has only magnitude (no direction). Examples include mass, temperature, and time.

Vector Notation

Vectors can be represented in several ways:

• Bold notation: $\mathbf{a}$, $\mathbf{b}$, $\mathbf{v}$
• Arrow notation: $\vec{AB}$, $\vec{v}$
• Underline notation: $\underline{a}$, $\underline{v}$

Vector components: A vector in 2D can be written as $\langle x, y \rangle$ or $x\mathbf{i} + y\mathbf{j}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors in the x and y directions respectively.

Types of Vectors

1. Zero Vector: Vector with magnitude 0, denoted as $\vec{0}$ or $\mathbf{0}$
2. Unit Vector: Vector with magnitude 1
3. Position Vector: Vector from origin to a point
4. Negative Vector: Vector with same magnitude but opposite direction

8.2 Addition and Subtraction of Vectors

Vector Addition

Vector addition follows the Triangle Law and Parallelogram Law.

Triangle Law: If $\vec{AB}$ and $\vec{BC}$ are two vectors, then $\vec{AC} = \vec{AB} + \vec{BC}$

Parallelogram Law: For vectors $\vec{a}$ and $\vec{b}$, the resultant $\vec{a} + \vec{b}$ is the diagonal of the parallelogram formed by $\vec{a}$ and $\vec{b}$

Component-wise Addition: If $\vec{a} = a_x\mathbf{i} + a_y\mathbf{j}$ and $\vec{b} = b_x\mathbf{i} + b_y\mathbf{j}$, then: $\vec{a} + \vec{b} = (a_x + b_x)\mathbf{i} + (a_y + b_y)\mathbf{j}$

Vector Subtraction

Vector subtraction is defined as addition of the negative vector: $\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$

Component-wise Subtraction: $\vec{a} - \vec{b} = (a_x - b_x)\mathbf{i} + (a_y - b_y)\mathbf{j}$

Properties of Vector Operations

1. Commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
2. Associative: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
3. Distributive: $k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}$ (where k is a scalar)

8.3 Vectors in a Cartesian Plane

Representation in Component Form

A vector $\vec{v}$ can be represented as:

• Column vector: $\begin{pmatrix} x \\ y \end{pmatrix}$
• Unit vector form: $x\mathbf{i} + y\mathbf{j}$
• Bracket notation: $\langle x, y \rangle$

Magnitude of a Vector

The magnitude (length) of vector $\vec{v} = x\mathbf{i} + y\mathbf{j}$ is: $|\vec{v}| = \sqrt{x^2 + y^2}$

Unit Vector

A unit vector in the direction of $\vec{v}$ is: $\hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{x\mathbf{i} + y\mathbf{j}}{\sqrt{x^2 + y^2}}$

Collinear Vectors

Vectors are collinear if they are parallel to each other (same or opposite direction).

Condition: Vectors $\vec{a}$ and $\vec{b}$ are collinear if $\vec{a} = k\vec{b}$ for some scalar k.

Important Formulas and Methods

Key Vector Operations

Vector Geometry

Position Vector: For point P(x,y), position vector is $\vec{OP} = x\mathbf{i} + y\mathbf{j}$

Vector between two points: $\vec{AB} = \vec{OB} - \vec{OA} = (x_B - x_A)\mathbf{i} + (y_B - y_A)\mathbf{j}$

Distance between points: $AB = |\vec{AB}| = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$

Solved Examples

Example 1: Basic Vector Operations

Given vectors $\vec{a} = 3\mathbf{i} + 2\mathbf{j}$ and $\vec{b} = \mathbf{i} - 4\mathbf{j}$, find: a) $\vec{a} + \vec{b}$ b) $\vec{a} - \vec{b}$ c) $2\vec{a} + 3\vec{b}$ d) $|\vec{a}|$

Solutions:

a) $\vec{a} + \vec{b} = (3\mathbf{i} + 2\mathbf{j}) + (\mathbf{i} - 4\mathbf{j}) = (3+1)\mathbf{i} + (2-4)\mathbf{j} = 4\mathbf{i} - 2\mathbf{j}$

b) $\vec{a} - \vec{b} = (3\mathbf{i} + 2\mathbf{j}) - (\mathbf{i} - 4\mathbf{j}) = (3-1)\mathbf{i} + (2+4)\mathbf{j} = 2\mathbf{i} + 6\mathbf{j}$

c) $2\vec{a} + 3\vec{b} = 2(3\mathbf{i} + 2\mathbf{j}) + 3(\mathbf{i} - 4\mathbf{j}) = 6\mathbf{i} + 4\mathbf{j} + 3\mathbf{i} - 12\mathbf{j} = 9\mathbf{i} - 8\mathbf{j}$

d) $|\vec{a}| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$

Example 2: Position Vectors and Vector Between Points

Given points A(2, 3) and B(5, 7), find: a) Position vectors $\vec{OA}$ and $\vec{OB}$ b) Vector $\vec{AB}$ c) Distance AB d) Unit vector in direction of $\vec{AB}$

Solutions:

a) $\vec{OA} = 2\mathbf{i} + 3\mathbf{j}$ $\vec{OB} = 5\mathbf{i} + 7\mathbf{j}$

b) $\vec{AB} = \vec{OB} - \vec{OA} = (5-2)\mathbf{i} + (7-3)\mathbf{j} = 3\mathbf{i} + 4\mathbf{j}$

c) $AB = |\vec{AB}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

d) Unit vector: $\hat{AB} = \frac{\vec{AB}}{|\vec{AB}|} = \frac{3\mathbf{i} + 4\mathbf{j}}{5} = \frac{3}{5}\mathbf{i} + \frac{4}{5}\mathbf{j}$

Example 3: Vector Addition Triangle Law

Use the triangle law to find the resultant of $\vec{PQ} = 2\mathbf{i} + 3\mathbf{j}$ and $\vec{QR} = -3\mathbf{i} + \mathbf{j}$. Also find the magnitude of the resultant.

Solution:

Using triangle law: $\vec{PR} = \vec{PQ} + \vec{QR} = (2\mathbf{i} + 3\mathbf{j}) + (-3\mathbf{i} + \mathbf{j}) = -\mathbf{i} + 4\mathbf{j}$

Magnitude: $|\vec{PR}| = \sqrt{(-1)^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17}$

Example 4: Unit Vector and Scalar Multiplication

Find a unit vector in the direction of $\vec{v} = 6\mathbf{i} - 8\mathbf{j}$. Also find the vector of magnitude 10 in the same direction.

Solution:

First, find magnitude: $|\vec{v}| = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Unit vector: $\hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{6\mathbf{i} - 8\mathbf{j}}{10} = \frac{3}{5}\mathbf{i} - \frac{4}{5}\mathbf{j}$

Vector of magnitude 10 in same direction: Since the unit vector has magnitude 1, multiply by 10: $10 \times (\frac{3}{5}\mathbf{i} - \frac{4}{5}\mathbf{j}) = 6\mathbf{i} - 8\mathbf{j}$ (which is the original vector)

Example 5: Collinear Vectors

Determine if vectors $\vec{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\vec{b} = 4\mathbf{i} + 6\mathbf{j}$ are collinear. If so, find the scalar k such that $\vec{a} = k\vec{b}$.

Solution:

Check if $\vec{a} = k\vec{b}$ for some scalar k: $2\mathbf{i} + 3\mathbf{j} = k(4\mathbf{i} + 6\mathbf{j}) = 4k\mathbf{i} + 6k\mathbf{j}$

This gives: 2 = 4k and 3 = 6k From first equation: k = 2/4 = 1/2 From second equation: k = 3/6 = 1/2

Since k is the same for both components, the vectors are collinear with k = 1/2.

Therefore: $\vec{a} = \frac{1}{2}\vec{b}$

Mathematical Derivations

Vector Addition Parallelogram Law

The parallelogram law states that if two vectors $\vec{a}$ and $\vec{b}$ are represented as adjacent sides of a parallelogram, then the diagonal from the common starting point represents their sum $\vec{a} + \vec{b}$.

Proof: Using component addition, if $\vec{a} = a_x\mathbf{i} + a_y\mathbf{j}$ and $\vec{b} = b_x\mathbf{i} + b_y\mathbf{j}$, then $\vec{a} + \vec{b} = (a_x + b_x)\mathbf{i} + (a_y + b_y)\mathbf{j}$. This corresponds to the diagonal of the parallelogram formed by the vectors.

Magnitude Formula Derivation

For a vector $\vec{v} = x\mathbf{i} + y\mathbf{j}$, the magnitude is the distance from origin to the point (x,y).

Using the Pythagorean theorem: $|\vec{v}| = \sqrt{x^2 + y^2}$

Unit Vector Formula

A unit vector has magnitude 1. To find the unit vector in the direction of $\vec{v}$, we divide $\vec{v}$ by its magnitude:

$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$

Since $|\hat{v}| = \left|\frac{\vec{v}}{|\vec{v}|}\right| = \frac{|\vec{v}|}{|\vec{v}|} = 1$, this gives a unit vector in the same direction.

Real-World Applications

1. Physics - Force and Motion

Vectors are fundamental in physics:

• Force: $\vec{F} = F_x\mathbf{i} + F_y\mathbf{j}$ represents force components
• Velocity: $\vec{v} = v_x\mathbf{i} + v_y\mathbf{j}$ for motion in 2D
• Acceleration: $\vec{a} = a_x\mathbf{i} + a_y\mathbf{j}$ for changing velocity

Example: A boat moving with velocity $\vec{v} = 10\mathbf{i} + 5\mathbf{j}$ km/h. The river current is $\vec{c} = -2\mathbf{i} + 0\mathbf{j}$ km/h. Actual velocity: $\vec{v} + \vec{c} = 8\mathbf{i} + 5\mathbf{j}$ km/h.

2. Engineering - Structural Analysis

Vectors are used to:

• Calculate resultant forces on structures
• Determine stress and strain components
• Analyze truss systems
• Design load-bearing structures

3. Computer Graphics

Vector applications in graphics:

• Object positioning: Vectors for location and orientation
• Animation: Velocity vectors for smooth motion
• Collision detection: Vector calculations for intersections
• 3D transformations: Rotation, scaling, translation vectors

4. Navigation

Navigation uses vectors for:

• Course planning: Direction vectors for routes
• Wind correction: Wind vector compensation
• GPS positioning: Location vectors
• Velocity calculations: Speed and direction vectors

Complex Problem-Solving Techniques

Problem: Find the vector $\vec{v}$ such that $\vec{v} + 2\vec{a} = 3\vec{b}$, where $\vec{a} = \mathbf{i} + 2\mathbf{j}$ and $\vec{b} = 3\mathbf{i} - \mathbf{j}$.

Solution:

$\vec{v} + 2\vec{a} = 3\vec{b}$ $\vec{v} = 3\vec{b} - 2\vec{a}$ $\vec{v} = 3(3\mathbf{i} - \mathbf{j}) - 2(\mathbf{i} + 2\mathbf{j})$ $\vec{v} = 9\mathbf{i} - 3\mathbf{j} - 2\mathbf{i} - 4\mathbf{j}$ $\vec{v} = 7\mathbf{i} - 7\mathbf{j}$

Problem: Given points A(1,2), B(3,-1), C(5,4), find:

a) Vector $\vec{AB}$ b) Vector $\vec{AC}$ c) The vector $2\vec{AB} + 3\vec{AC}$ d) The magnitude of the result from c)

Solution:

a) $\vec{AB} = \vec{OB} - \vec{OA} = (3-1)\mathbf{i} + (-1-2)\mathbf{j} = 2\mathbf{i} - 3\mathbf{j}$

b) $\vec{AC} = \vec{OC} - \vec{OA} = (5-1)\mathbf{i} + (4-2)\mathbf{j} = 4\mathbf{i} + 2\mathbf{j}$

c) $2\vec{AB} + 3\vec{AC} = 2(2\mathbf{i} - 3\mathbf{j}) + 3(4\mathbf{i} + 2\mathbf{j}) = 4\mathbf{i} - 6\mathbf{j} + 12\mathbf{i} + 6\mathbf{j} = 16\mathbf{i} + 0\mathbf{j} = 16\mathbf{i}$

d) Magnitude: $|16\mathbf{i}| = \sqrt{16^2 + 0^2} = 16$

Problem: Find a unit vector perpendicular to both $\vec{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\vec{b} = \mathbf{i} - \mathbf{j}$.

Solution:

In 2D, a vector perpendicular to $\vec{a} = a_x\mathbf{i} + a_y\mathbf{j}$ is $\vec{a}_{\perp} = -a_y\mathbf{i} + a_x\mathbf{j}$

For $\vec{a} = 2\mathbf{i} + 3\mathbf{j}$, perpendicular vector is $-3\mathbf{i} + 2\mathbf{j}$

Check: $(2\mathbf{i} + 3\mathbf{j}) \cdot (-3\mathbf{i} + 2\mathbf{j}) = 2(-3) + 3(2) = -6 + 6 = 0$ ✓

Magnitude: $|-3\mathbf{i} + 2\mathbf{j}| = \sqrt{(-3)^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}$

Unit perpendicular vector: $\frac{-3\mathbf{i} + 2\mathbf{j}}{\sqrt{13}} = -\frac{3}{\sqrt{13}}\mathbf{i} + \frac{2}{\sqrt{13}}\mathbf{j}$

Summary Points

• Vectors have both magnitude and direction
• Vector addition follows triangle and parallelogram laws
• Vectors can be represented in component form: $x\mathbf{i} + y\mathbf{j}$
• The magnitude of vector $\vec{v} = x\mathbf{i} + y\mathbf{j}$ is $\sqrt{x^2 + y^2}$
• Unit vectors have magnitude 1 and are found by dividing by magnitude
• Vectors are collinear if one is a scalar multiple of another
• Vector operations follow commutative, associative, and distributive properties

Common Mistakes to Avoid

1. Confusing vectors and scalars - Remember vectors have direction, scalars don't
2. Magnitude calculation errors - Always use Pythagorean theorem correctly
3. Unit vector confusion - Divide by magnitude, not multiply
4. Collinearity errors - Check if vectors are scalar multiples
5. Component addition errors - Add corresponding components separately

SPM Exam Tips

Exam Strategies

1. Master notation - Be comfortable with different vector notations
2. Practice component operations - Addition, subtraction, scalar multiplication
3. Understand geometric interpretations - Triangle law, parallelogram law
4. Calculate magnitudes accurately - Use Pythagorean theorem carefully
5. Show all working - Vector problems often have multiple steps

Key Exam Topics

• Vector notation and representation (15% of questions)
• Vector addition and subtraction (25% of questions)
• Magnitude calculations (20% of questions)
• Unit vectors (20% of questions)
• Collinear vectors and applications (20% of questions)

Time Management Tips

• Basic vector operations: 3-4 minutes
• Magnitude and unit vector problems: 4-5 minutes
• Collinearity problems: 3-4 minutes
• Complex applications: 6-8 minutes

Practice Problems

Level 1: Basic Vector Operations

1. Given $\vec{a} = 2\mathbf{i} - 3\mathbf{j}$ and $\vec{b} = \mathbf{i} + 4\mathbf{j}$, find: a) $\vec{a} + \vec{b}$ b) $\vec{a} - \vec{b}$ c) $3\vec{a} + 2\vec{b}$ d) $|\vec{a}|$

2. Calculate the magnitude of each vector: a) $3\mathbf{i} + 4\mathbf{j}$ b) $-2\mathbf{i} + 5\mathbf{j}$ c) $6\mathbf{i} - 8\mathbf{j}$ d) $\mathbf{i} + \mathbf{j}$

Level 2: Position Vectors

3. Given points A(3,1) and B(7,-2), find: a) Position vectors $\vec{OA}$ and $\vec{OB}$ b) Vector $\vec{AB}$ c) Distance AB d) Unit vector in direction of $\vec{AB}$

4. Find the vector from P(-2,4) to Q(3,-1) and its magnitude.

Level 3: Unit Vectors

5. Find unit vectors in the direction of: a) $3\mathbf{i} + 4\mathbf{j}$ b) $-6\mathbf{i} + 8\mathbf{j}$ c) $\mathbf{i} - \mathbf{j}$ d) $2\mathbf{i} - 2\mathbf{j}$

6. Find vectors of magnitude 8 in the direction of: a) $2\mathbf{i} + 2\mathbf{j}$ b) $3\mathbf{i} - 4\mathbf{j}$ c) $-\mathbf{i} + \mathbf{j}$

Level 4: Collinear Vectors

7. Determine if the following vectors are collinear. If yes, find the scalar k: a) $\vec{a} = 4\mathbf{i} + 6\mathbf{j}$, $\vec{b} = 2\mathbf{i} + 3\mathbf{j}$ b) $\vec{a} = \mathbf{i} - 2\mathbf{j}$, $\vec{b} = 3\mathbf{i} + 4\mathbf{j}$ c) $\vec{a} = -3\mathbf{i} + 6\mathbf{j}$, $\vec{b} = 1.5\mathbf{i} - 3\mathbf{j}$

Level 5: Applications

8. Physics: A boat moves with velocity $\vec{v} = 12\mathbf{i} + 5\mathbf{j}$ km/h in still water. The river current is $\vec{c} = -3\mathbf{i} + 2\mathbf{j}$ km/h. Find the boat's actual velocity.

9. Engineering: Two forces act on a point: $\vec{F_1} = 20\mathbf{i} + 15\mathbf{j}$ N and $\vec{F_2} = -10\mathbf{i} + 25\mathbf{j}$ N. Find the resultant force and its magnitude.

10. Navigation: An airplane has velocity $\vec{v} = 200\mathbf{i} + 100\mathbf{j}$ km/h relative to air. Wind velocity is $\vec{w} = -20\mathbf{i} + 10\mathbf{j}$ km/h. Find ground velocity.

Did You Know? 📚

Vectors were first systematically studied by Irish mathematician William Rowan Hamilton in the 19th century. He developed quaternions, a generalization of complex numbers to four dimensions, which laid the foundation for modern vector analysis. The modern vector notation we use today was developed by J. Willard Gibbs and Oliver Heaviside in the late 19th century, simplifying Hamilton's complex quaternions into the more manageable vector form used today.

Quick Reference Guide

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Vectors provide a powerful mathematical language for describing quantities with direction and magnitude. Mastering vector operations is essential for advanced studies in physics, engineering, and computer science, and forms the foundation for understanding more complex mathematical concepts.