Cambridge (CIE) IGCSE — Additional Maths
Vectors in Two Dimensions
The big idea: A vector is just a set of instructions for a journey — "go this far, in this direction." Once you can add those journeys together, scale them up or down, and work with the coordinates (positions) they connect, you can solve almost any problem about shapes, distances, or moving objects.
Summary — What This Chapter Covers
- A vector has both magnitude (size) and direction — unlike a scalar, which is just a number.
- Vectors can be written as column vectors
(x, y) or using unit vectors i and j as xi + yj.
- The magnitude (length) of a vector is found using Pythagoras' theorem: |v| = √(x² + y²).
- A unit vector has magnitude 1 — you find one by dividing a vector by its own magnitude.
- Position vectors describe where a point is relative to the origin O. Displacement vectors describe the journey between two points.
- To add or subtract vectors, add or subtract their matching components (top with top, bottom with bottom).
- Multiplying a vector by a scalar changes its size (and flips its direction if the scalar is negative) but keeps it parallel to the original.
- Vectors are used to solve geometry problems: proving lines are parallel, finding midpoints, proving points are collinear, and working with ratios on a line.
- Vectors can represent velocity too — and position at time t is found using r = r₀ + vt.
1. Introduction to Vectors
What Actually Is a Vector?
Think of a vector as a set of travel directions, not a place. "Walk 3 steps east and 7 steps south" is a vector — it tells you a magnitude (how far) and a direction (which way). Compare that to a scalar, which is just a plain number with no direction attached — like "it's 20°C outside" or "the bag weighs 5 kg."
A classic example from the chapter: two objects moving at 7 m/s and −7 m/s are travelling at the same speed (7 m/s each) but in opposite directions. Speed doesn't care about direction — velocity does. That's the difference between a scalar and a vector in a nutshell.
Vectors don't just appear in "arrow on a grid" questions — you've already met them when translating graphs (shifting a curve left/right and up/down is literally applying a vector), and they show up constantly in mechanics to model forces, displacement, velocity, and acceleration.
How Vectors Are Written
There are two main notations you need to be fluent switching between:
Column vector: i, j (unit vector) form:
→ ( x ) →
AB = ( y ) AB = xi + yj
Example: → ( 3 )
AB = (-7) = 3i - 7j
"3 units right, 7 units down"
If you know the start point (A) and end point (B), you can write the vector as AB with an arrow drawn above it, showing it goes specifically from A to B. Note that AB and BA are not the same — they point in opposite directions, so BA = -AB.
Vectors can also be written using a single bold lowercase letter, like a or b. Since you can't write in bold in an exam, you should underline it instead: a.
Examiner Tip
Think of every vector as a journey from one place to another. If a diagram isn't given in the question, draw one yourself — it doesn't need to be perfectly accurate, it just needs to show the relationships between vectors clearly.
Magnitude of a Vector