Mathematics Form 4 | Vectors (2) | Masomotele Free Learning

📋 Learning Outcomes

By the end of this lesson, you should be able to:

Express vectors in column notation and component form
Find the magnitude and direction of a vector
Add, subtract and multiply vectors by scalars
Use position vectors to describe points in a plane
Apply the section formula (dividing a line in a given ratio)
Prove geometric properties using vectors
Apply vectors to solve real-life navigation problems

🔍 Key Inquiry Question: How can vectors be used to describe movement, position and geometric relationships precisely?

📐 Vector Notation and Magnitude

Vector Representation

A vector has both magnitude (size) and direction.

Column vector: a = (x, y) where x = horizontal component, y = vertical component

|a| = √(x² + y²) (magnitude/modulus)

Example:

a = (3, 4): |a| = √(9+16) = √25 = 5

b = (−5, 12): |b| = √(25+144) = √169 = 13

Unit Vector

â = a / |a|

Unit vector has magnitude = 1

Example:

Find unit vector of a = (3, 4). |a| = 5

â = (3/5, 4/5) = (0.6, 0.8)

➕ Vector Operations

Addition and Subtraction

a = (3,2), b = (1,−4)

a + b = (3+1, 2+(−4)) = (4, −2)

a − b = (3−1, 2−(−4)) = (2, 6)

Scalar Multiplication

k·a = k(x, y) = (kx, ky)

3a = 3(3,2) = (9, 6)

−2b = −2(1,−4) = (−2, 8)

Parallel Vectors

Vectors are parallel if one is a scalar multiple of the other: a = kb for some scalar k

Example:

Are (4, 6) and (6, 9) parallel?

6/4 = 1.5 and 9/6 = 1.5 → ratio equal → Yes, parallel (k = 1.5)

📍 Position Vectors

Definition

The position vector of point P(x,y) relative to origin O is OP = (x, y).

Vector AB = OB − OA = b − a (position vector of B minus position vector of A)

Example:

A(2,3), B(5,7). Find AB and |AB|.

AB = OB − OA = (5,7) − (2,3) = (3, 4)

|AB| = √(9+16) = 5 units

Midpoint of AB

M = ½(a + b) = ((x₁+x₂)/2, (y₁+y₂)/2)

Example:

Midpoint of A(2,3) and B(6,7): M = (4, 5)

✂️ Section Formula

Dividing a Line in a Given Ratio

If P divides AB in ratio m:n internally:

OP = (na + mb) / (m + n)

Example 1:

A(1,2), B(7,8). P divides AB in ratio 2:1. Find OP.

OP = (1×(1,2) + 2×(7,8)) / (2+1) = ((1,2)+(14,16))/3 = (15,18)/3 = (5, 6)

Example 2 — Midpoint check:

Ratio 1:1 → OP = (a+b)/2 = midpoint ✓

External Division

OP = (mb − na) / (m − n) (m ≠ n)

📐 Geometric Proofs Using Vectors

Key Vector Facts for Proofs

If AB = kCD, then AB is parallel to CD
If AB = AC then B = C (same endpoint)
Collinear points: AB = kAC (scalar multiple)

Worked Proof

Example: Prove midpoints form a parallelogram

Let ABCD be a quadrilateral. P, Q, R, S are midpoints of AB, BC, CD, DA.

PQ = AQ − AP = ½AC (using midpoint theorem)

SR = AR − AS = ½AC

Since PQ = SR, PQRS is a parallelogram ✓

Collinearity Proof

Example:

OA = 2i + 3j, OB = 4i + 6j. Show A, O, B are collinear.

OB = 2(2i+3j) = 2·OA → OB is a scalar multiple of OA

Therefore A, O, B are collinear ✓

🧭 Navigation Problems

Bearing and Velocity Vectors

Example 1: Resultant displacement

A ship sails 30 km East then 40 km North. Find resultant displacement.

d₁ = (30, 0), d₂ = (0, 40)

Resultant = (30, 40), |d| = √(900+1600) = 50 km

Direction = tan⁻¹(40/30) = 53.1° from East = N36.9°E

Example 2: Relative velocity

Wind velocity = (10, 5) km/h, plane velocity = (200, 0) km/h

Resultant = (210, 5) km/h

Speed = √(210²+5²) ≈ 210.06 km/h

📊 Vector Summary

Key Formulas

Concept	Formula
Magnitude	\|a\| = √(x²+y²)
Unit vector	â = a/\|a\|
Vector AB	AB = OB − OA
Midpoint	M = ½(a+b)
Section (m:n)	P = (na+mb)/(m+n)
Parallel	a = kb
Collinear	AB = kAC

✏️ Practice Problems

Problem 1

a = (4,3), b = (−1,2). Find: a+b, 2a−3b, |a|, unit vector of b.

Solution:

a+b = (3,5) | 2a−3b = (8,6)−(−3,6) = (11,0)

|a| = √(16+9) = 5 | |b| = √5, b̂ = (−1/√5, 2/√5)

Problem 2

P(1,2) and Q(7,5). Find: PQ, midpoint M, point R that divides PQ in ratio 2:1.

Solution:

PQ = (6,3) | M = (4, 3.5)

R = (1×(1,2) + 2×(7,5))/3 = ((1,2)+(14,10))/3 = (5, 4)

📝 Quiz — Vectors (2)

Part A: Fill in the Blanks (5 marks)

1. The magnitude of vector (5, 12) is .

2. Vector AB = OB OA.

3. Two vectors are parallel if one is a multiple of the other.

4. The midpoint of A(2,4) and B(6,8) is .

5. A unit vector has magnitude equal to .

Part B: Short Answer (10 marks)

6. (4 marks) OA = (2,3), OB = (8,9). Find AB, |AB|, midpoint M of AB, and the point P that divides AB in ratio 1:2.

7. (3 marks) Show that A(1,2), B(3,5), C(7,11) are collinear.

8. (3 marks) A plane flies with velocity (300,0) km/h. Wind blows at (−20,50) km/h. Find the resultant velocity and speed.