1. Introduction

In physics and engineering, some quantities are described completely just by their magnitude (a number), while others need both magnitude and direction to be fully described. This chapter builds the mathematics for handling the second type.

• Scalar quantities: Have only magnitude. Examples: mass, distance, speed, time, temperature.
• Vector quantities: Have both magnitude and direction. Examples: displacement, velocity, force, acceleration.

2. Basic Definitions

3. Types of Vectors

Type	Definition
Zero vector (Null vector)	A vector with magnitude zero, initial and terminal points coincide. Denoted 0.
Unit vector	A vector with magnitude 1. Unit vector along a is â = a / |a|.
Coinitial vectors	Two or more vectors having the same initial point.
Collinear vectors	Vectors that are parallel to the same line, regardless of magnitude or direction.
Equal vectors	Vectors having the same magnitude and same direction.
Negative of a vector	A vector with the same magnitude but opposite direction, e.g. −a.

4. Algebra of Vectors

(a) Addition of Vectors

Triangle Law: If two vectors are represented in magnitude and direction by two sides of a triangle taken in order, their sum (resultant) is represented by the third side taken in the opposite order.

AB + BC = AC

Parallelogram Law: If two vectors are represented by two adjacent sides of a parallelogram, their sum is represented by the diagonal of the parallelogram passing through their common point.

Properties of vector addition:

• Commutative: a + b = b + a
• Associative: (a + b) + c = a + (b + c)
• Additive identity: a + 0 = a
• Additive inverse: a + (−a) = 0

(b) Multiplication of a Vector by a Scalar

If a is a vector and λ is a scalar, then λa is a vector whose magnitude is |λ||a|, in the same direction as a if λ > 0, and opposite direction if λ < 0.

(c) Position Vector: Section Formula

If P divides the line segment joining A(a) and B(b) internally in the ratio m : n, then position vector of P is:

r = (m·b + n·a) / (m + n)

For the midpoint (m = n = 1):

r = (a + b) / 2

5. Components of a Vector

Let i, j, k be unit vectors along the x, y, z axes. Any vector r with components x, y, z can be written as:

r = xi + yj + zk

Magnitude: |r| = √(x² + y² + z²)

If A(x₁, y₁, z₁) and B(x₂, y₂, z₂) are two points, then:

AB = (x₂−x₁)i + (y₂−y₁)j + (z₂−z₁)k

|AB| = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

6. Product of Two Vectors

(a) Scalar (Dot) Product

The scalar product of two nonzero vectors a and b is defined as:

a · b = |a||b| cos θ

where θ is the angle between a and b (0 ≤ θ ≤ π).

Properties:

• a · b = b · a (commutative)
• a · (b + c) = a · b + a · c (distributive)
• a · a = |a|²
• i · i = j · j = k · k = 1
• i · j = j · k = k · i = 0
• If a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then a · b = a₁b₁ + a₂b₂ + a₃b₃

Angle between two vectors:

cos θ = (a · b) / (|a||b|)

Projection of a on b:

Projection = (a · b) / |b|

💡 Note: If a · b = 0 and neither a nor b is the zero vector, then a and b are perpendicular (orthogonal).

(b) Vector (Cross) Product

The vector product of two nonzero, non-parallel vectors a and b is a vector whose magnitude is:

|a × b| = |a||b| sin θ

and whose direction is perpendicular to both a and b, given by the right-hand rule.

Properties:

• a × b = −(b × a) (not commutative)
• a × (b + c) = a × b + a × c (distributive)
• i × i = j × j = k × k = 0
• i × j = k, j × k = i, k × i = j

Determinant formula: If a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then:

a × b = determinant of the matrix with rows [i, j, k], [a₁, a₂, a₃], [b₁, b₂, b₃]

Applications:

• If a × b = 0 and a, b are nonzero, then a and b are parallel (or collinear).
• Area of a triangle with adjacent sides a and b: (1/2)|a × b|
• Area of a parallelogram with adjacent sides a and b: |a × b|
• Area of a parallelogram with diagonals d₁ and d₂: (1/2)|d₁ × d₂|

7. Important Formula Summary

Concept	Formula
Unit vector along a	â = a / |a|
Section formula (internal)	r = (mb + na)/(m+n)
Dot product	a·b = |a||b|cosθ = a₁b₁+a₂b₂+a₃b₃
Cross product magnitude	|a×b| = |a||b|sinθ
Angle between vectors	cosθ = (a·b)/(|a||b|)
Area of triangle	(1/2)|a×b|
Area of parallelogram	|a×b|

8. Solved Examples

9. Quick Revision Points

• Vectors have magnitude and direction; scalars have only magnitude.
• Dot product gives a scalar; cross product gives a vector.
• a · b = 0 → vectors are perpendicular.
• a × b = 0 → vectors are parallel.
• Always remember i·i=j·j=k·k=1 and i·j=j·k=k·i=0 for dot product shortcuts.
• Always remember i×i=j×j=k×k=0 and the cyclic rule i×j=k, j×k=i, k×i=j for cross product shortcuts.