Free GCSE Maths lesson: Geometry and Measures
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Lesson 52 · GCSE / Key Stage 4 · Maths · Geometry and Measures

Vectors

Learn how vectors describe movement with size and direction, then use vector journeys to solve GCSE geometry questions.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Geometry and Measures

Lesson overview

Vectors describe a journey using direction and distance.

GCSE vector questions ask you to combine arrows, scale arrows and explain geometric facts such as parallel lines or points on a straight line.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandGeometry and Measures
Tier Higher
Calculator status Calculator and non-calculator
Exam-board status Higher-tier geometry skill

Visual summary

Before viewing

Before viewing, trace the horizontal and vertical movement shown by one vector.

Infographic explaining GCSE Maths vectors, including column vectors, vector addition, scalar multiples, position vectors, proof checks and sign errors.
Use this visual to read vector direction, add matching components, spot scalar multiples and write a clear proof reason.Download visual

After viewing

After viewing, write one route as a vector expression and simplify it.

What you will learn

  • How to read column vectors and directed line segments.
  • How to add and subtract vectors by following journeys.
  • How to multiply vectors by numbers.
  • How to write routes through diagrams in vector notation.
  • How to use position vectors, midpoints and ratio points.
  • How to use parallel and equal vectors in short proofs.

Key facts before you start

Vector A vector has magnitude and direction.
Column vector A column vector shows horizontal and vertical movement.
Parallel vectors Parallel vectors are scalar multiples.
Route notation Equivalent vector routes give the same movement.

Why this matters

Vectors are useful whenever direction matters: map movement, forces, velocity, translations and proof in geometry. At GCSE, they help you turn a diagram into exact algebra.

Prior knowledge

You should already be comfortable with:

  • using coordinates on a grid,
  • adding and subtracting positive and negative numbers,
  • recognising parallel lines,
  • using ratio and scale factors,
  • writing clear algebraic working.

Clear explanation

Column vectors show horizontal and vertical movement

A column vector tells you how far to move sideways and how far to move up or down. For example, 32 means move 3 units right and 2 units up. A negative number means move left or down.

Vector journeys can be added

If one journey is a and the next journey is b, the whole journey is a + b. Reversing a vector changes its sign, so the reverse of a is −a.

Vector journeys on a grid A grid shows vector a moving three squares right and one square up, vector b moving one square right and two squares up, and the combined vector a plus b moving four squares right and three squares up. P a b a + b Each square is 1 unit. Read the arrows a = (3, 1) b = (1, 2) a + b = (4, 3) right/left first, then up/down.
Checked diagram: vector a moves 3 right and 1 up; vector b moves 1 right and 2 up; together they move 4 right and 3 up.

Scalar multiples show parallel vectors

A scalar is an ordinary multiplier. If b = 2a, then b is twice as long as a in the same direction. If b = −2a, it is twice as long in the opposite direction. Non-zero scalar multiples are parallel.

For proof, the route matters as much as the final expression. State the journey you are taking, simplify carefully, then write a sentence such as "these vectors are scalar multiples, so the lines are parallel" or "the equal vectors meet at B, so the points are collinear".

Position-vector questions use the origin as the starting point. If point A has position vector a and point B has position vector b, then AB = b − a. Midpoints and ratio points are weighted averages, so keep the order of the endpoints clear before simplifying.

Worked examples

Example 1: Add column vectors

Add the vectors ( 4 , −1 ) and ( −2 , 5 ).

Horizontal: 4 + (−2) = 2 Vertical: −1 + 5 = 4
Reveal answer

Answer: ( 2 , 4 ).

Example 2: Find a route

In a parallelogram, AB = a and AD = b. Find AC.

A to C = A to B + B to C BC is equal to AD, so BC = b
Reveal answer

Answer: AC = a + b.

Example 3: Parallel vectors

Vector u = ( 2 , 3 ). Explain why v = ( 6 , 9 ) is parallel to u.

v = 3u
Reveal answer

Answer: v is a scalar multiple of u, so the vectors are parallel.

Example 4: A point in a ratio

Point P has position vector p and point Q has position vector q. R is one quarter of the way from P to Q. Write the position vector of R.

PR = 14(q − p) OR = OP + PR = p + 14(q − p)
Reveal answer

Answer: 34p + 14q.

Quick checks

Choose an answer, then check your thinking.

1. What does the vector ( −3 , 4 ) mean?

2. If a = ( 2 , 1 ), what is 3a?

3. Which statement proves two non-zero vectors are parallel?

Practice questions

Question 1

Write the column vector for a movement 5 units right and 2 units down.

Reveal answer and marking guidance

Answer: ( 5 , −2 ).

Marking: Put the horizontal movement first and use a negative vertical component for down.

Question 2

Add ( 7 , 3 ) and ( −4 , 6 ).

Reveal answer and marking guidance

Answer: ( 3 , 9 ).

Marking: Add matching components: 7 + (−4) = 3 and 3 + 6 = 9.

Question 3

If a = ( −2 , 5 ), find −2a.

Reveal answer and marking guidance

Answer: ( 4 , −10 ).

Marking: Multiply both components by −2.

Question 4

In a parallelogram, AB = p and AD = q. Write vector BD in terms of p and q.

Reveal answer and marking guidance

Answer: −p + q, or q − p.

Marking: Travel from B to A using −p, then from A to D using q.

Question 5

Are ( 3 , −6 ) and ( −1 , 2 ) parallel? Explain your answer.

Reveal answer and marking guidance

Answer: yes, because ( 3 , −6 ) = −3( −1 , 2 ).

Marking: Show one vector is a scalar multiple of the other.

Question 6

Point A has position vector a and point B has position vector b. M is the midpoint of AB. Write the position vector of M.

Reveal answer and marking guidance

Answer: (a + b) ÷ 2.

Marking: Average the two position vectors because M is halfway between A and B.

Question 7

Point A has position vector a and point B has position vector b. Write vector AB in terms of a and b.

Reveal answer and marking guidance

Answer: b − a.

Marking: Travel from the origin to A in reverse, then from the origin to B: AB = −a + b = b − a.

Question 8

Points P, Q and R have position vectors p, q and 2q − p. Show that P, Q and R are collinear.

Reveal answer and marking guidance

Answer: PQ = q − p and QR = (2q − p) − q = q − p, so PQ = QR. Therefore P, Q and R are collinear.

Marking: Find two directed line segments and state that equal vectors sharing Q lie on the same straight line.

Question 9

Point A has position vector a and point B has position vector b. Point C is two thirds of the way from A to B. Write the position vector of C.

Reveal answer and marking guidance

Answer: 13a + 23b.

Marking: Start at A and add two thirds of AB: a + 23(b − a) = 13a + 23b.

Question 10

Vector u = 6−4 and vector v = −32. Explain why the two vectors are parallel.

Reveal answer and marking guidance

Answer: They are parallel because u = −2v.

Marking: Multiply v by −2: −2 × −32 = 6−4. A scalar multiple shows the vectors are parallel, with opposite directions because the multiplier is negative.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For vector questions, marks usually come from reading direction correctly, adding matching components, using negative signs when an arrow is reversed, multiplying every component by a scalar, handling position vectors in the right order, and writing a short reason when proving vectors are parallel or collinear.

Common mistakes

  • Swapping the components: read horizontal movement first, then vertical movement.
  • Forgetting negative directions: left and down are negative in column vectors.
  • Only multiplying one component: a scalar multiple changes every part of the vector.
  • Using the wrong arrow direction: AB and BA are opposites.
  • Claiming parallel without a reason: show one vector is a scalar multiple of the other.

Extension challenge

Points A, B and C have position vectors a, b and 2b − a. Prove that A, B and C lie on a straight line.

Reveal answer

Answer: AB = b − a and BC = (2b − a) − b = b − a, so AB and BC are equal vectors. Therefore A, B and C lie on the same straight line with B between A and C.

Exam-board guidance

Vectors are a GCSE Maths geometry skill across the supported boards, most commonly on extension papers. Questions may use column vectors, labelled diagram vectors, midpoints, ratios, parallel lines or short proof.

AQA GCSE Maths

Focus on the shared method, notation and checking habit; wording and context can vary by route.

OCR GCSE Maths

Show clear method steps and use precise notation; question wording may vary by route.

Pearson Edexcel GCSE Maths

Keep working visible and check units or notation; contexts may vary by route.

Eduqas GCSE Maths

Use the same core skill in practical contexts, with clear reasoning and final checks.

WJEC Wales

Connect the method to the context and state what the result means.

CCEA GCSE Maths

Make each method step visible and keep notation, units and final checks clear.

Next lesson

Next, move on to Circle Theorems.

Previous lesson: Congruence and Similarity Back to GCSE Maths lessons Next lesson: Circle Theorems