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A2 Level · Pure Mathematics 4 WMA14

Vectors in 2D

Revise Vectors in 2D for Pure Mathematics 4 WMA14 (A2 Level) — revision notes and instant AI marking.

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What You'll Learn

What vectors are and why they matter in maths
How to represent vectors in different forms
Finding magnitude (size) and direction of vectors
Adding, subtracting, and multiplying vectors
Unit vectors and their special property
Position vectors vs displacement vectors
Using vectors to prove lines are parallel
Finding distance between two points
Showing points are collinear (on the same line)
Dividing line segments in a given ratio

1. What is a Vector?

The Big Picture

A vector is a quantity that has both magnitude (size) and direction. Unlike a scalar (which is just a number like "5 metres"), a vector tells you both how far to go AND which way.

Think of it like this: If someone asks "How do I get from A to B?", saying "50 metres" is useless. You need to say "50 metres north" — that's a vector. The direction is just as important as the distance.

In maths, we use vectors to describe translations, forces, velocities, and displacements. You've probably already seen them when translating function graphs.

How to Write Vectors

Vectors can be written in several equivalent ways. All three below represent the exact same vector:

Column Vector (most common in exams): AB = ( 3 ) ( -2 ) i, j Notation (unit vectors): AB = 3i − 2j Bold or Underlined: AB or AB (in handwriting, use underlining)

Column Vector: Written as a vertical fraction with an x-component (top) and y-component (bottom). The top number tells you how far left/right, the bottom tells you how far up/down.

i, j Notation: A way of writing vectors using unit vectors. i is a unit vector pointing right (1 unit), and j is a unit vector pointing up (1 unit). So 3i − 2j means "go 3 units right and 2 units down".

Understanding the Arrow

When we write AB, the arrow goes from A to B. This matters:

AB starts at A and ends at B
BA starts at B and ends at A — it's the opposite direction
So BA = −AB

Practice: Converting Between Forms

Question 1

A vector has an x-component of 4 and a y-component of −7. Write this vector in:

(a) Column vector form

(b) i, j notation

2. Magnitude and Direction

What is Magnitude?

The magnitude of a vector is its length — how far you travel if you follow the vector. It's always positive (or zero).

If a vector is written as a = (x, y) or a = xi + yj, then we find its magnitude using Pythagoras' theorem:

|a| = √(x² + y²)

Why Pythagoras? If you move x units right and y units up, you've traced out a right-angled triangle. The vector itself is the hypotenuse. So yes, Pythagoras applies.

Example: Finding Magnitude

3. Adding and Subtracting Vectors

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Also in the full note

4. Scalar Multiplication and Parallel Vectors
5. Position Vectors
6. Proving Points Are Collinear
7. Dividing a Line Segment in a Given Ratio
8. Using Vectors to Prove Geometric Properties
Key Concepts to Memorise
Concepts Checklist
Exam Tips and Common Mistakes

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Vectors in 2D

Vectors in 2D

What You'll Learn

1. What is a Vector?

The Big Picture

How to Write Vectors

Understanding the Arrow

Practice: Converting Between Forms

Question 1

2. Magnitude and Direction

What is Magnitude?

Example: Finding Magnitude

3. Adding and Subtracting Vectors

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