IGCSE MATH: VECTORS

Vectors in Two Dimensions - Study Notes

Vectors in Two Dimensions: Complete Guide

E7.2 - Vectors in Two Dimensions

What is a Vector?

A vector is a quantity that has both:

Magnitude (size/length)
Direction (which way it points)

Examples: displacement, velocity, force

Vector Notation

Vectors can be written in several ways:

Bold letters: a, AB
Underlined: a̲, AB̲
With arrow: AB→ (read as "vector AB")
Column vector: (x/y) where x is horizontal, y is vertical

Example notations for the same vector:

AB→
a
(3/4) - meaning 3 units right, 4 units up

1. Describing Translations Using Vectors

A vector describes how to move from one point to another:

The top number tells you horizontal movement (x-direction)
The bottom number tells you vertical movement (y-direction)

Column Vector Form:

a = (x/y) means:
• Move x units horizontally (+ right, - left)
• Move y units vertically (+ up, - down)

Example 1: Reading a Vector

Vector: a = (5/3)

Meaning: Move 5 units right and 3 units up

Vector: b = (-2/4)

Meaning: Move 2 units left and 4 units up

Vector: c = (3/-5)

Meaning: Move 3 units right and 5 units down

2. Adding and Subtracting Vectors

Addition Rule:
(a/b) + (c/d) = (a+c / b+d)

Add corresponding components separately

Subtraction Rule:
(a/b) - (c/d) = (a-c / b-d)

Subtract corresponding components separately

Example 2: Vector Addition

Question: Find a + b where a = (3/2) and b = (1/4)

Solution:

a + b = (3/2) + (1/4)
= (3+1 / 2+4)
= (4/6)

Answer: (4/6)

Meaning: Combined movement is 4 right, 6 up

Example 3: Vector Subtraction

Question: Find a - b where a = (5/3) and b = (2/1)

Solution:

a - b = (5/3) - (2/1)
= (5-2 / 3-1)
= (3/2)

Answer: (3/2)

Geometric Meaning:
• a + b: Follow vector a, then follow vector b
• a - b: The vector from the end of b to the end of a

3. Multiplying a Vector by a Scalar

A scalar is just a number (not a vector)

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

Scalar Multiplication Rule:

k × (a/b) = (ka / kb)

Multiply each component by the scalar k

Example 4: Scalar Multiplication

Question: Find 3a where a = (2/-1)

Solution:

3a = 3 × (2/-1)
= (3×2 / 3×(-1))
= (6/-3)

Answer: (6/-3)

Meaning: Three times as long in the same direction

Example 5: Negative Scalar

Question: Find -2b where b = (3/4)

Solution:

-2b = -2 × (3/4)
= (-2×3 / -2×4)
= (-6/-8)

Answer: (-6/-8)

Meaning: Twice as long in the OPPOSITE direction

Important Facts:

Multiplying by a positive number: same direction, different length
Multiplying by a negative number: opposite direction
Multiplying by 1: no change
Multiplying by 0: gives zero vector (0/0)

E7.3 - Magnitude of a Vector

What is Magnitude?

The magnitude of a vector is its length or size

It's always a positive number (or zero)

Denoted by: |a| or |AB→|

Magnitude Formula:

For vector a = (x/y):

|a| = √(x² + y²)

This comes from Pythagoras' theorem!

Example 6: Finding Magnitude

Question: Find |a| where a = (3/4)

Solution:

|a| = √(3² + 4²)
= √(9 + 16)
= √25
= 5

Answer: 5 units

Example 7: Magnitude with Decimals

Question: Find |b| where b = (5/2)

Solution:

|b| = √(5² + 2²)
= √(25 + 4)
= √29
≈ 5.39 units

Answer: √29 or 5.39 units (to 2 d.p.)

Example 8: Negative Components

Question: Find |c| where c = (-6/-8)

Solution:

|c| = √((-6)² + (-8)²)
= √(36 + 64)
= √100
= 10

Answer: 10 units

Note: Negative components become positive when squared!

E7.4 - Vector Geometry

1. Representing Vectors by Directed Line Segments

Vectors can be drawn as arrows on a grid:

The length of the arrow represents magnitude
The direction of the arrow shows direction
The starting point can be anywhere (vectors are about movement, not position)

Key Idea: Two vectors are equal if they have the same magnitude and direction, even if they start at different points!

2. Position Vectors

A position vector describes the position of a point relative to the origin (0, 0)

If point A has coordinates (x, y), its position vector is OA→ = (x/y)

Example 9: Position Vectors

Question: Point A is at (3, 5). What is the position vector OA→?

Solution:

Position vector from origin O(0,0) to A(3,5)
OA→ = (3/5)

Answer: (3/5)

3. Expressing Vectors in Terms of Two Coplanar Vectors

Coplanar vectors lie in the same plane (2D space)

Any 2D vector can be expressed as a combination of two non-parallel vectors

These are called base vectors or component vectors

Standard Unit Vectors:

i = (1/0) → one unit right
j = (0/1) → one unit up

Any vector (a/b) = ai + bj

Example 10: Expressing in Terms of i and j

Question: Express v = (4/3) in terms of i and j

Solution:

v = (4/3)
= 4i + 3j

Answer: 4i + 3j

Example 11: Sum and Difference of Vectors

Question: In triangle ABC, AB→ = p and BC→ = q. Express AC→ in terms of p and q

Solution:

To go from A to C, we can go A→B→C
AC→ = AB→ + BC→
AC→ = p + q

Answer: p + q

4. Using Vectors to Solve Geometric Problems

Applications Include:

Showing vectors are parallel
Showing points are collinear (lie on same line)
Solving problems with ratio and similarity

Parallel Vectors:
Two vectors are parallel if one is a scalar multiple of the other
If a = kb (where k is a number), then a and b are parallel

Example 12: Showing Vectors are Parallel

Question: Show that a = (6/9) and b = (2/3) are parallel

Solution:

Check if a = kb for some number k
(6/9) = k(2/3)
6 = 2k → k = 3
9 = 3k → k = 3 ✓
Since a = 3b, they are parallel

Answer: Vectors are parallel (a = 3b)

Collinear Points:
Three points A, B, C are collinear if AB→ and AC→ are parallel
This means AB→ = k × AC→ for some scalar k

Example 13: Showing Points are Collinear

Question: Points A, B, C have position vectors (1/2), (3/5), (5/8). Are they collinear?

Solution:

Find AB→ = (3/5) - (1/2) = (2/3)
Find AC→ = (5/8) - (1/2) = (4/6)
Check if parallel: AC→ = (4/6) = 2(2/3) = 2 × AB→
Since AC→ = 2AB→, the vectors are parallel
They share point A, so A, B, C are collinear

Answer: Yes, points are collinear

Example 14: Ratio and Similarity

Question: Point M divides AB in ratio 2:3. If OA→ = a and OB→ = b, find OM→

Solution:

M divides AB in ratio 2:3 means AM:MB = 2:3
Total parts = 2 + 3 = 5
M is 2/5 of the way from A to B
OM→ = OA→ + (2/5)AB→
AB→ = OB→ - OA→ = b - a
OM→ = a + (2/5)(b - a)
= a + (2/5)b - (2/5)a
= (3/5)a + (2/5)b

Answer: OM→ = (3/5)a + (2/5)b

Common Mistakes to Avoid

❌ Confusing magnitude with direction
❌ Forgetting to square the components when finding magnitude
❌ Adding/subtracting x and y components together (keep separate!)
❌ Forgetting that magnitude is always positive
❌ Not checking if vectors are parallel when proving collinearity
❌ Mixing up position vectors with displacement vectors

Practice Problems

Vector Operations:

If a = (3/4) and b = (2/-1), find:
- a) a + b
- b) a - b
- c) 2a + 3b

Magnitude:

Find the magnitude of:
- a) (5/12)
- b) (-3/4)
- c) (7/-24)

Parallel Vectors:

Are vectors (4/6) and (6/9) parallel? Show your working.

Collinearity:

Points P, Q, R have position vectors (2/3), (5/7), (8/11). Are they collinear?

Answers:

a) (5/3) b) (1/5) c) (12/5)
a) 13 b) 5 c) 25
Yes, (6/9) = 1.5 × (4/6), so they are parallel
Yes, PQ→ = (3/4) and PR→ = (6/8) = 2(3/4), so collinear

Quick Reference Summary

Operation	Formula	Example
Addition	(a/b) + (c/d) = (a+c / b+d)	(2/3) + (1/4) = (3/7)
Subtraction	(a/b) - (c/d) = (a-c / b-d)	(5/6) - (2/1) = (3/5)
Scalar multiplication	k(a/b) = (ka/kb)	3(2/4) = (6/12)
Magnitude	\|(a/b)\| = √(a² + b²)	\|(3/4)\| = √(9+16) = 5
Parallel test	a = kb	(6/9) = 3(2/3) → parallel