Vectors
Column vectors, vector addition and subtraction, scalar multiplication, magnitude, position vectors, and vector geometry proofs.
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The Arrow of Direction
You have mastered trigonometry — finding distances and angles in triangles. Now let us learn a completely new way to describe movement.
Rule
A vector is a quantity that has both size AND direction. Speed tells you how fast, but velocity tells you how fast and which way. Distance tells you how far, but displacement tells you how far and in what direction.
In this topic, you will learn to write vectors as columns of numbers, add and subtract them, scale them, and use them to prove geometric facts about shapes. Vectors are the language of physics and engineering.
| Scalars (plain numbers) | Vectors (arrows) |
| Only have size — e.g. 5 km, 30°, 8 seconds | Have size AND direction — e.g. 5 km north, 30 m/s east, a force of 8 N downward |
What You Will Learn
You will start with column vectors — two numbers stacked vertically that describe a sliding movement. Then you will learn to add vectors (chaining journeys), multiply by scalars (stretching or shrinking), and find their magnitude (length).
Steps
- 1
Column vectors: describe movement with two numbers.
- 2
Vector arithmetic: add, subtract, and scale vectors.
- 3
Position vectors: every point gets a unique vector from the origin.
- 4
Vector proofs: show lines are parallel, points are collinear, or midpoints divide segments.
Formula
The workhorse formula:
This finds the vector between any two points using their position vectors.
Practise vectors now
Interactive visual aids, then real past-paper questions marked on your working like the exam. Free to start.
Practise free on LemmaraLemmara is an independent study tool, not affiliated with or endorsed by Cambridge International Education or Pearson Edexcel. The lessons and worked examples are Lemmara’s own.