Vector Algebra – AAI ATC Exam Prep

📘 About This Chapter

Vector Algebra is a crucial and high-weightage chapter in AAI ATC Mathematics. This chapter covers Types of Vectors, Addition, Scalar Multiplication, Dot Product, Cross Product and their geometric applications — all frequently tested in AAI ATC written exams.

Scalars & Vectors Types of Vectors Vector Addition Dot Product Cross Product Section Formula

Basic Concepts & Types of Vectors

A scalar has only magnitude (e.g., mass, temperature, speed). A vector has both magnitude and direction (e.g., force, velocity, displacement).

📌 Key Vector Types

Zero Vector (0̄): Initial and terminal points coincide. Magnitude = 0.

Unit Vector (â): Magnitude = 1. â = ā/|ā|

Coinitial: Vectors with the same initial point.

Collinear: Vectors parallel to the same line (same or opposite direction).

Equal Vectors: Same magnitude AND direction.

Negative Vector: Same magnitude, opposite direction. BA̅ = −AB̅

Position Vector: For point P(x, y, z) w.r.t. origin O:

OP̄ (or r̄) = xî + yĵ + zk̂ and |OP̄| = √(x² + y² + z²)

Direction Cosines: If α, β, γ are angles made with x, y, z axes:

cos α = x/r = l, cos β = y/r = m, cos γ = z/r = n

l² + m² + n² = 1 (important identity!)

🖼️ Image Space
Add a 3D coordinate diagram showing a vector OP with direction angles α, β, γ labelled, and the position vector components

Practice MCQs – Basic Concepts & Types

Q1. Which of the following is a scalar quantity?

✅ Answer: C — Work done = Force × displacement × cosθ. It has only magnitude (no direction), so it is a scalar. Force, velocity, and displacement are vectors.

Q2. The direction cosines of the vector r̄ = î + ĵ − 2k̂ are:

✅ Answer: C — |r̄| = √(1+1+4) = √6. So l = 1/√6, m = 1/√6, n = −2/√6. Verify: l²+m²+n² = 1/6+1/6+4/6 = 1 ✓

Q3. The position vector of a point P is 3î + 4ĵ. The magnitude |OP| is:

✅ Answer: C — |OP| = √(3² + 4²) = √(9+16) = √25 = 5 units.

Vector Addition & Scalar Multiplication

Vector addition has two key laws. When vectors are added in a triangle or parallelogram, the rules follow naturally from displacement geometry.

📌 Laws of Vector Addition

Triangle Law: AC̄ = AB̄ + BC̄ (the resultant is the closing side)

Parallelogram Law: If ā and b̄ are adjacent sides of a parallelogram, their sum ā + b̄ is the diagonal through the common point.

Properties: Commutative: ā + b̄ = b̄ + ā | Associative: (ā+b̄)+c̄ = ā+(b̄+c̄)

Scalar Multiplication: If ā is a vector and λ is a scalar:

|λā| = |λ| × |ā| (magnitude scales by |λ|)

Unit vector: â = ā / |ā|

Component Form Operations: If ā = a₁î + a₂ĵ + a₃k̂ and b̄ = b₁î + b₂ĵ + b₃k̂:

ā + b̄ = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂

λā = (λa₁)î + (λa₂)ĵ + (λa₃)k̂

🖼️ Image Space
Add a diagram showing Triangle Law and Parallelogram Law of vector addition side by side with arrows labelled ā, b̄ and ā+b̄

Collinearity Condition: b̄ = λā for some nonzero scalar λ ⟺ b₁/a₁ = b₂/a₂ = b₃/a₃

Practice MCQs – Vector Addition & Scalar Multiplication

Q1. If ā = 2î − 3ĵ + k̂ and b̄ = î + 4ĵ − 2k̂, then |ā + b̄| is:

✅ Answer: C — ā + b̄ = (2+1)î + (−3+4)ĵ + (1−2)k̂ = 3î + ĵ − k̂. |ā+b̄| = √(9+1+1) = √11. (Select the closest option — the computed answer is √11.)

Q2. The unit vector in the direction of ā = 2î + 3ĵ + k̂ is:

✅ Answer: B — |ā| = √(4+9+1) = √14. Unit vector â = ā/|ā| = (2î+3ĵ+k̂)/√14.

Q3. Which condition confirms that vectors 2î − 3ĵ + 4k̂ and 4î − 6ĵ + 8k̂ are collinear?

✅ Answer: B — 4/2 = −6/−3 = 8/4 = 2. So b̄ = 2ā, confirming collinearity. Two vectors are collinear iff one is a scalar multiple of the other.

Section Formula & Vector Components

Any vector r̄ = xî + yĵ + zk̂ can be decomposed into components along coordinate axes. The scalars x, y, z are called scalar components (or rectangular components).

📌 Section Formula

If R divides PQ (with position vectors ā and b̄) in the ratio m:n:

Internal Division: OR̄ = (mb̄ + nā) / (m+n)

External Division: OR̄ = (mb̄ − nā) / (m−n)

Midpoint (m=n): OR̄ = (ā + b̄) / 2

Vector joining two points P₁(x₁,y₁,z₁) and P₂(x₂,y₂,z₂):

P₁P₂̄ = (x₂−x₁)î + (y₂−y₁)ĵ + (z₂−z₁)k̂

|P₁P₂̄| = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]

🖼️ Image Space
Add a diagram showing internal and external division of segment PQ by point R in ratio m:n with vectors labelled

Example: If P has position vector 3ā − 2b̄ and Q has ā + b̄, find R dividing PQ internally in ratio 2:1.

OR̄ = [2(ā+b̄) + 1(3ā−2b̄)] / (2+1) = (5ā) / 3 = 5ā/3

Practice MCQs – Section Formula & Components

Q1. The position vector of the midpoint of the segment joining A(2î − ĵ + k̂) and B(4î + ĵ − 3k̂) is:

✅ Answer: B — Midpoint = (ā+b̄)/2 = [(2î−ĵ+k̂)+(4î+ĵ−3k̂)]/2 = (6î+0ĵ−2k̂)/2 = 3î+0ĵ−k̂.

Q2. The vector joining P(2,3,0) to Q(−1,−2,−4) directed from P to Q is:

✅ Answer: B — PQ̄ = Q−P = (−1−2)î+(−2−3)ĵ+(−4−0)k̂ = −3î − 5ĵ − 4k̂.

Q3. Point R divides the join of P(ā) and Q(b̄) externally in ratio 2:1. Position vector of R is:

✅ Answer: C — External division formula: OR̄ = (mb̄−nā)/(m−n) = (2b̄−1·ā)/(2−1) = 2b̄ − ā.

Scalar (Dot) Product of Two Vectors

The dot product of two nonzero vectors ā and b̄ is a scalar. It encodes the angle between them.

📌 Definition & Key Results

ā · b̄ = |ā| |b̄| cos θ (where θ is angle between them)

Perpendicular vectors: ā · b̄ = 0 (when θ = 90°)

Parallel vectors: ā · b̄ = |ā||b̄| (when θ = 0°)

Component formula: ā · b̄ = a₁b₁ + a₂b₂ + a₃b₃

Unit vector identities: î·î = ĵ·ĵ = k̂·k̂ = 1 | î·ĵ = ĵ·k̂ = k̂·î = 0

Angle between two vectors:

cos θ = (ā · b̄) / (|ā| |b̄|) = (a₁b₁ + a₂b₂ + a₃b₃) / (|ā||b̄|)

Projection of ā on b̄:

Projection = (ā · b̄) / |b̄| = (1/|b̄|)(ā · b̄)

Important Identity:

|ā − b̄|² = |ā|² − 2(ā·b̄) + |b̄|²

🖼️ Image Space
Add a diagram showing two vectors ā and b̄ with angle θ between them, and the projection of ā on b̄ labelled

Example: Find angle between ā = î+ĵ−k̂ and b̄ = î−ĵ+k̂

ā · b̄ = (1)(1)+(1)(−1)+(−1)(1) = 1−1−1 = −1

|ā| = |b̄| = √3 → cos θ = −1/3 → θ = cos⁻¹(−1/3)

Practice MCQs – Scalar (Dot) Product

Q1. If ā = 2î + 3ĵ + 2k̂ and b̄ = î + 2ĵ + k̂, then ā · b̄ is:

✅ Answer: B — ā·b̄ = (2)(1)+(3)(2)+(2)(1) = 2+6+2 = 10.

Q2. The projection of vector ā = 2î + 3ĵ + 2k̂ on vector b̄ = î + 2ĵ + k̂ is:

✅ Answer: C — Projection = (ā·b̄)/|b̄| = 10/√(1+4+1) = 10/√6.

Q3. If |ā| = 2, |b̄| = 3, and ā·b̄ = 4, then |ā − b̄| is:

✅ Answer: A — |ā−b̄|² = |ā|²−2(ā·b̄)+|b̄|² = 4−8+9 = 5. So |ā−b̄| = √5.

Vector (Cross) Product of Two Vectors

The cross product of two nonzero vectors ā and b̄ is a vector perpendicular to both. It uses the right-hand rule to determine direction.

📌 Definition & Key Results

ā × b̄ = |ā| |b̄| sinθ · n̂ (n̂ ⊥ both ā and b̄)

Parallel vectors: ā × b̄ = 0̄ (when θ = 0° or 180°)

Perpendicular vectors: |ā × b̄| = |ā||b̄| (when θ = 90°)

NOT commutative: ā × b̄ = −(b̄ × ā)

Unit vector identities: î×î = ĵ×ĵ = k̂×k̂ = 0̄

î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ

Determinant Formula:

ā × b̄ = |î ĵ k̂| |a₁ a₂ a₃| |b₁ b₂ b₃| = (a₂b₃−a₃b₂)î − (a₁b₃−a₃b₁)ĵ + (a₁b₂−a₂b₁)k̂

Area Applications:

Area of Triangle = ½|ā × b̄| (where ā, b̄ are two sides)

Area of Parallelogram = |ā × b̄| (where ā, b̄ are adjacent sides)

🖼️ Image Space
Add a 3D diagram showing ā × b̄ = n̂ using the right-hand rule, with angle θ between ā and b̄ and the resulting perpendicular vector

Example: Find |ā × b̄| where ā = 2î+ĵ+3k̂ and b̄ = 3î+5ĵ−2k̂

ā × b̄ = î(1·(−2)−3·5) − ĵ(2·(−2)−3·3) + k̂(2·5−1·3)

= î(−2−15) − ĵ(−4−9) + k̂(10−3) = −17î + 13ĵ + 7k̂

|ā × b̄| = √(289+169+49) = √507

Practice MCQs – Vector (Cross) Product

Q1. If ā = î and b̄ = ĵ, then ā × b̄ is:

✅ Answer: C — î × ĵ = k̂. This is one of the fundamental cross product identities for unit vectors (right-hand rule: i→j→k).

Q2. The area of a triangle with adjacent sides ā = 3î + ĵ + 4k̂ and b̄ = î − ĵ + k̂ is:

✅ Answer: B — ā×b̄ = î(1+4)−ĵ(3−4)+k̂(−3−1) = 5î+ĵ−4k̂. |ā×b̄| = √(25+1+16) = √42. Area = ½√42.

Q3. If ā and b̄ are unit vectors and the angle between them is π/4, then |ā × b̄| is:

✅ Answer: C — |ā×b̄| = |ā||b̄|sinθ = 1×1×sin(π/4) = 1/√2.

Important Identities & Quick Revision

These identities are directly asked in AAI ATC exams — memorise them!

Concept	Formula / Result	Notes
Position Vector	r̄ = xî+yĵ+zk̂	\|r̄\| = √(x²+y²+z²)
Direction Cosines	l²+m²+n² = 1	l=x/r, m=y/r, n=z/r
Unit Vector	â = ā/\|ā\|	\|â\| = 1 always
Dot Product	ā·b̄ = \|ā\|\|b̄\|cosθ	Scalar result
Perpendicularity	ā·b̄ = 0	θ = 90°
Cross Product	\|ā×b̄\| = \|ā\|\|b̄\|sinθ	Vector result
Parallelism	ā×b̄ = 0̄	θ = 0° or 180°
Triangle Area	½\|ā×b̄\|	ā,b̄ = two sides
Parallelogram Area	\|ā×b̄\|	ā,b̄ = adjacent sides
Cauchy-Schwartz	\|ā·b̄\| ≤ \|ā\|\|b̄\|	Equality when parallel

⚠️ AAI ATC Exam Watch Points

1. Cross product is NOT commutative (ā×b̄ = −b̄×ā) but dot product IS commutative.

2. î·î = ĵ·ĵ = k̂·k̂ = 1 | î×î = ĵ×ĵ = k̂×k̂ = 0̄

3. If ā·b̄ = 0 AND ā×b̄ = 0̄, then at least one of ā or b̄ must be zero vector.

4. |ā+b̄|² = |ā|² + 2ā·b̄ + |b̄|² — frequently used in problems.

🖼️ Image Space
Add a summary mind-map or cheat sheet image showing all major formulas of Vector Algebra at a glance

Practice MCQs – Identities & Applications

Q1. If ā+b̄+c̄ = 0 and |ā|=3, |b̄|=4, |c̄|=5, then ā·b̄ + b̄·c̄ + c̄·ā is:

✅ Answer: B — |ā+b̄+c̄|²=0 → |ā|²+|b̄|²+|c̄|²+2(ā·b̄+b̄·c̄+c̄·ā)=0 → 9+16+25+2μ=0 → μ=−25.

Q2. The value of î·(ĵ×k̂) + ĵ·(î×k̂) + k̂·(î×ĵ) is:

✅ Answer: C — ĵ×k̂=î, so î·î=1. î×k̂=−ĵ, so ĵ·(−ĵ)=−1. î×ĵ=k̂, so k̂·k̂=1. Sum = 1−1+1 = 1.

Q3. λā is a unit vector (|ā| = a, λ ≠ 0 scalar) if:

✅ Answer: D — |λā| = |λ|·|ā| = |λ|·a = 1. So a = 1/|λ|.

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Chapter 10: Vector Algebra