Notation, operations, magnitude, and proof questions — everything you need to master vectors at GCSE.
Vectors are one of the most misunderstood topics in GCSE Higher Maths, and a lot of that misunderstanding comes from notation. Once you're clear on what the notation means and why it works, the actual calculations are quite straightforward. The proof questions at the end of a vectors question are where most students lose marks — this guide covers exactly how to tackle those too.
A vector is a quantity with both magnitude (size) and direction. This is different from a scalar, which has magnitude only. Speed is a scalar. Velocity is a vector. Distance is a scalar. Displacement is a vector.
At GCSE, vectors are used to describe movement between points. The vector from point A to point B is written as AB with an arrow above it, or in bold as a in textbooks. In handwriting, underline the letter: a.
Vectors can be written in column vector form, showing horizontal and vertical components:
The top number is the horizontal movement (positive = right, negative = left). The bottom number is the vertical movement (positive = up, negative = down).
In exam questions, vectors are usually given as letters: a, b, p, q etc. You manipulate these letters algebraically without needing to know their actual numerical values — which is where students sometimes get confused. You're doing algebra with vectors, not arithmetic.
To add vectors, add the components. Geometrically, this means following one vector and then the other — the resultant vector is the direct route from start to finish.
In terms of journeys: if going from A to B gives vector a, and going from B to C gives vector b, then going directly from A to C gives vector a + b.
Subtracting a vector is the same as adding its negative. The negative of a vector reverses its direction.
Geometrically: if AB = a, then BA = −a. This is a crucial fact for writing vector paths — if you need to go backwards along a known vector, you negate it.
Multiplying a vector by a scalar (a regular number) changes its magnitude but not its direction (unless the scalar is negative, which reverses direction).
This is important for proof questions. If one vector is a scalar multiple of another, the two vectors are parallel. This is the key fact behind almost every "show that two lines are parallel" question.
If vector PQ = k × vector RS for any scalar k, then PQ and RS are parallel. This is the single most important fact for vectors proof questions.
Given a diagram with several points and some known vectors, you often need to find the vector between two other points. The method is to find a path between those points using the vectors you know, then add them up.
OABC is a parallelogram where OA = a and OC = c. M is the midpoint of AB. Find the vector OM.
In a parallelogram, CB is parallel and equal to OA, so CB = a.
Path from O to M: go O → C → B → M.
OC = c
CB = a (same as OA, since OABC is a parallelogram)
BM = −½a (M is the midpoint of AB, so BM is half of BA, and BA = −a)
OM = OC + CB + BM = c + a − ½a = c + ½a
When finding a vector between two points, trace any route between them using vectors you know. You can go via as many intermediate points as you need — as long as every step uses a known vector (or the negative of one, or a scalar multiple). Then collect like terms at the end.
The magnitude (length) of a vector is found using Pythagoras' theorem. For a vector with components (x, y):
Example: vector a = (5, 12). |a| = √(25 + 144) = √169 = 13.
Note the vertical bar notation — |a| means "the magnitude of vector a". Don't confuse this with absolute value of a number, though the idea is similar.
The hardest part of any vectors question at GCSE is usually a "show that" or "prove that" part at the end. These ask you to prove geometrical facts using vector algebra — typically that two lines are parallel, or that three points are collinear (lie on the same straight line).
Find the vector expression for each line. If one is a scalar multiple of the other, they are parallel. State this explicitly — don't just show the algebra and assume the examiner knows why it means parallel.
Example: Show that PQ is parallel to RS given that PQ = 2a + 4b and RS = a + 2b.
PQ = 2(a + 2b) = 2 × RS
Since PQ = 2RS, PQ is a scalar multiple of RS, therefore PQ is parallel to RS.
Three points P, Q, R are collinear if the vector PQ is a scalar multiple of PR (or QR — any two vectors between the three points will do). If they are scalar multiples AND share a common point, the three points must lie on the same straight line.
⚠️ For collinearity, you must state TWO things: (1) the vectors are parallel (scalar multiples), AND (2) they share a common point. Without both, you haven't proved collinearity — just that the lines are parallel to each other, which could mean they're separate parallel lines.
OAB is a triangle. OA = a, OB = b. P is the midpoint of OA. Q is on AB such that AQ:QB = 2:1. Show that O, P and Q are NOT collinear, or find the vector OQ.
OP = ½a (P is midpoint of OA)
OQ = OA + AQ = a + ⅔(b − a) = a + ⅔b − ⅔a = ⅓a + ⅔b
OP = ½a. OQ = ⅓a + ⅔b. OQ is not a scalar multiple of OP (it contains a b component), so O, P and Q are not collinear.
The full vectors specification for each exam board — including exactly what proof skills are tested — is in the AQA GCSE Maths specification and the OCR GCSE Maths specification.
PaperPlus generates vectors questions at GCSE Higher level — including proof questions — with full mark schemes for AQA, Edexcel and OCR. Completely free.
Start Practising Free →Going the wrong direction along a vector. If you need to travel from B to A but only know vector AB, you must use −a, not a. Drawing an arrow on your diagram and labelling direction explicitly prevents this error.
Not collecting like terms. After tracing a path and adding vectors, always simplify by collecting terms in each letter. An answer of a + b + a − ½b should be simplified to 2a + ½b before you write it as your final answer.
Incomplete proofs. In proof questions, showing the algebra is not enough — you must state the conclusion explicitly. "Therefore PQ is parallel to RS" or "therefore O, P, Q are collinear" must appear in your working.
Confusing position vectors with displacement vectors. A position vector gives a point's location relative to the origin. A displacement vector describes movement between two points. In most GCSE questions they're interchangeable, but be careful when a question specifically uses the word "position vector".