Comprehensive Vector Notes

Comprehensive Guide to Vectors

1. Vector Basics

Vectors are mathematical objects that have both magnitude (size) and direction. They are used to represent physical quantities like velocity, acceleration, force, and momentum.

1.1 Vector Notation

Vectors can be written in various ways:

Bold letter: v
Letter with arrow: v or v̅
Component form: v = (v₁, v₂, v₃) or v = v₁i + v₂j + v₃k

1.2 Vector Representation

A 2D vector can be represented as an ordered pair (x, y), while a 3D vector can be represented as (x, y, z).

Example: Position Vector

The position vector of point P(3, 4) relative to the origin is v = (3, 4) or v = 3i + 4j. Its magnitude is |v| = √(3² + 4²) = 5 units.

2. Vector Operations

2.1 Vector Addition

To add vectors, add their corresponding components:

v + w = (v₁, v₂, v₃) + (w₁, w₂, w₃) = (v₁+w₁, v₂+w₂, v₃+w₃)

Example: Vector Addition

Given v = (3, 1, 2) and w = (4, -2, 5):

v + w = (3, 1, 2) + (4, -2, 5) = (7, -1, 7)

2.2 Vector Subtraction

To subtract vectors, subtract their corresponding components:

v - w = (v₁, v₂, v₃) - (w₁, w₂, w₃) = (v₁-w₁, v₂-w₂, v₃-w₃)

Example: Vector Subtraction

Given v = (5, 8, 3) and w = (2, 3, 1):

v - w = (5, 8, 3) - (2, 3, 1) = (3, 5, 2)

2.3 Scalar Multiplication

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative):

kv = k(v₁, v₂, v₃) = (kv₁, kv₂, kv₃)

Example: Scalar Multiplication

Given v = (3, -2, 4) and k = 2:

kv = 2(3, -2, 4) = (6, -4, 8)

This doubles the magnitude of v while keeping the same direction.

3. Vector Products

3.1 Dot Product (Scalar Product)

The dot product of two vectors results in a scalar value:

v · w = v₁w₁ + v₂w₂ + v₃w₃ = |v||w|cos(θ)

where θ is the angle between the vectors

Properties of Dot Product:

If v · w = 0, then v and w are perpendicular (orthogonal)
v · v = |v|²
v · w = w · v (commutative)

Example: Dot Product

Given v = (2, 3, -1) and w = (4, 0, 5):

v · w = (2)(4) + (3)(0) + (-1)(5) = 8 + 0 - 5 = 3

Finding the Angle Between Vectors

We can find the angle between two vectors using the dot product formula:

cos(θ) = (v · w) / (|v||w|)

θ = cos⁻¹((v · w) / (|v||w|))

Example: Finding Angle Between Vectors

Given v = (1, 1, 0) and w = (0, 1, 1):

v · w = (1)(0) + (1)(1) + (0)(1) = 0 + 1 + 0 = 1

|v| = √(1² + 1² + 0²) = √2

|w| = √(0² + 1² + 1²) = √2

cos(θ) = (v · w) / (|v||w|) = 1 / (√2 × √2) = 1/2

θ = cos⁻¹(1/2) = 60°

3.2 Cross Product (Vector Product)

The cross product of two vectors results in a vector that is perpendicular to both original vectors:

v × w = (v₂w₃ - v₃w₂, v₃w₁ - v₁w₃, v₁w₂ - v₂w₁)

|v × w| = |v||w|sin(θ)

where θ is the angle between the vectors

Properties of Cross Product:

v × w = -(w × v) (anti-commutative)
v × v = 0
The magnitude of v × w equals the area of the parallelogram formed by v and w
The right-hand rule determines the direction of v × w

Example: Cross Product

Given v = (2, 3, 4) and w = (5, 6, 7):

v × w = ((3)(7) - (4)(6), (4)(5) - (2)(7), (2)(6) - (3)(5))

v × w = (21 - 24, 20 - 14, 12 - 15)

v × w = (-3, 6, -3)

4. Vector Properties

4.1 Magnitude (Length)

The magnitude or length of a vector is calculated using the Pythagorean theorem:

|v| = √(v₁² + v₂² + v₃²)

Example: Vector Magnitude

Given v = (3, 4, 12):

|v| = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13

4.2 Unit Vector

A unit vector has a magnitude of 1 and maintains the same direction as the original vector:

v̂ = v/|v|

Example: Unit Vector

Given v = (0, 4, 3):

|v| = √(0² + 4² + 3²) = √(0 + 16 + 9) = √25 = 5

v̂ = v/|v| = (0, 4, 3)/5 = (0, 4/5, 3/5)

4.3 Direction Cosines

Direction cosines are the cosines of the angles between a vector and the coordinate axes:

cos(α) = v₁/|v|, cos(β) = v₂/|v|, cos(γ) = v₃/|v|

where α, β, γ are the angles with the x, y, and z axes respectively

Example: Direction Cosines

Given v = (2, 2, 1):

|v| = √(2² + 2² + 1²) = √(4 + 4 + 1) = √9 = 3

cos(α) = 2/3, cos(β) = 2/3, cos(γ) = 1/3

α ≈ 48.2°, β ≈ 48.2°, γ ≈ 70.5°

5. Vector Applications and Problem Types

5.1 Projection of Vectors

The projection of vector v onto vector w is given by:

proj_w v = (v · w / |w|²) × w

The scalar projection is: |proj_w v| = |v · w| / |w|

Example: Vector Projection

Given v = (4, 5, 0) and w = (2, 0, 0):

v · w = (4)(2) + (5)(0) + (0)(0) = 8

|w|² = 2² = 4

proj_w v = (8/4) × (2, 0, 0) = 2 × (2, 0, 0) = (4, 0, 0)

5.2 Work Done by a Force

Work is the dot product of force and displacement vectors:

W = F · d = |F||d|cos(θ)

Example: Work Calculation

A force F = (3, 4, 0) N moves an object through a displacement d = (5, 2, 0) m:

W = F · d = (3)(5) + (4)(2) + (0)(0) = 15 + 8 + 0 = 23 Joules

5.3 Torque

Torque is the cross product of position and force vectors:

τ = r × F

Example: Torque Calculation

A force F = (0, 5, 0) N is applied at position r = (2, 0, 0) m relative to the axis of rotation:

τ = r × F = (2, 0, 0) × (0, 5, 0) = (0, 0, 10) N·m

6. Methods for Solving Vector Problems

Problem Type	Method	Key Formulas
Finding Vector Magnitude	Apply Pythagorean theorem to the components	\|v\| = √(v₁² + v₂² + v₃²)
Finding Unit Vector	Divide the vector by its magnitude	v̂ = v/\|v\|
Finding Angle Between Vectors	Use the dot product formula	cos(θ) = (v · w) / (\|v\|\|w\|)
Testing for Orthogonality	Check if dot product equals zero	v · w = 0
Finding Vector Projection	Use the projection formula	proj_w v = (v · w / \|w\|²) × w
Finding a Perpendicular Vector	In 2D: swap components and negate one In 3D: use cross product with another vector	For 2D: v = (a, b) → v⊥ = (-b, a) For 3D: v⊥ = v × w
Decomposing a Vector	Break into components along desired directions	v = v₁i + v₂j + v₃k

Strategy for Vector Problems

Identify the vector quantities involved (position, velocity, force, etc.)
Determine the appropriate vector operation (addition, dot product, cross product, etc.)
Express vectors in component form to simplify calculations
Apply the relevant formulas and solve algebraically
Interpret the result in the context of the problem