GCSE Vectors — A Complete Guide (2D) | The Maths and Science Tutor

Vectors are the topic that Higher tier students often leave to last — they look unfamiliar, the notation is new, and the proof-style questions at the end can feel slippery. They shouldn't. The arithmetic is genuinely the easiest in the entire syllabus once you see the pattern. The only difficulty is in the geometry, and that comes from practice. This guide walks through everything you need for GCSE Higher.

What a vector is

A vector has two things: magnitude (length) and direction. We usually write a vector as a column:

a = (3 over 4) — meaning 3 right and 4 up

The top number is the horizontal component (positive = right, negative = left). The bottom is the vertical component (positive = up, negative = down).

Adding and subtracting vectors

Add vectors by adding their components separately. Same for subtraction.

Worked example 1

If a = (3, 4) and b = (1, −2), find a + b and a − b.

a + b = (3 + 1, 4 + (−2)) = (4, 2)

a − b = (3 − 1, 4 − (−2)) = (2, 6)

Scalar multiplication

Multiplying a vector by a number (a "scalar") multiplies both components.

Worked example 2

If a = (2, 5), find 3a and −a.

3a = (6, 15)

−a = (−2, −5)

Geometrically: 3a points in the same direction as a but is three times as long. −a points in the opposite direction with the same length.

Magnitude (length) of a vector

Use Pythagoras on the components.

Worked example 3

Find the magnitude of a = (5, 12).

|a| = √(5² + 12²) = √(25 + 144) = √169 = 13

Vector geometry — the Higher proof questions

This is where most marks live on the topic. You'll get a diagram with vectors named on each side, then asked to find expressions for other vectors and prove that two lines are parallel or that three points are collinear (lie on the same straight line).

Worked example 4 — typical exam-style

OAB is a triangle with O the origin. OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.

Travel from O to M by going O → A → M (halfway from A to B).

OM = OA + ½AB

AB = AO + OB = −a + b = b − a

So OM = a + ½(b − a) = a + ½b − ½a = ½a + ½b = ½(a + b)

Examiner's note: The proof questions almost always come down to writing a single vector two different ways and showing they match (parallel) or are scalar multiples (collinear). If two vectors are scalar multiples of each other and they share a point, the three relevant points are collinear.

The mistakes that cost the most marks

Mistake 1 — Direction errors. AB and BA point in opposite directions. BA = −AB. Half the proof-question marks are lost to careless direction signs.

Mistake 2 — Not factorising the answer. If you arrive at 3a + 6b and the question is asking for proof of parallelism, factor it as 3(a + 2b) — that scalar 3 is what proves the parallel claim. Examiners want to see the factor explicitly.

Mistake 3 — Confusing a vector with its magnitude. a = (3, 4) is the vector itself; |a| = 5 is its length. They're different objects with different uses.

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Vectors — A Complete 2D Guide