Write vectors along routes in terms of a and b, then prove lines are parallel or that three points lie on a straight line. It is Higher only, aimed at grade 8. Free questions with answers are on this page.
Vector proof shows that midpoints, parallelograms and collinear points do what they should. Position vectors from a common origin: AB=b−a. To prove ABCD is a parallelogram, show AB=DC (or the diagonals bisect: 2a+c=2b+d). Collinear: one vector is a scalar multiple of the other, and they share a point.
Keep everything in terms of the given letters a,b. A midpoint M of AB is 2a+b. Finish with a sentence: 'so PQ=2MN, hence PQ is parallel to MN and twice as long'.
Write every vector in terms of the given letters. Midpoint: OM=21(a+b). Parallel: one vector is a scalar multiple of the other. Collinear: two vectors sharing a point are scalar multiples. Factorise to make the multiple obvious.
Objectives
Write a midpoint as 1/2(a+b).
Spot a scalar multiple, which means parallel.
Keep the answer as a vector, not a single coordinate when both matter.
Key points
Write a vector between two points by walking a route through known vectors, e.g. AB=−a+b.
Use ratios and midpoints to find fractions of a vector, e.g. AP:PB =2:1 means AP=32AB.
Simplify to the form pa+qb and factorise out a common number.
Prove two vectors are parallel by showing one is a scalar multiple of the other.
Worked examples
Example 1
OA = a and OB = b. M is the midpoint of AB. Write the position vector OM.
2 marks
Average the two position vectors
OM = 1/2 (a + b)
1/2(a+b)
Answer: 1/2(a+b)
Common slips
Slip: Add them and stop A midpoint is halfway.
Slip: OM = a + b The half is required.
Slip: a+b That is the position of M.
Example 2
AB = (2, 4) and AC = (6, 12). Find the scalar k such that AC = k AB.
2 marks
Compare components
6 / 2 = 3 and 12 / 4 = 3
3
Answer: 3
Common slips
Slip: Add the vectors A scalar multiplies both components.
Slip: 6 - 2 = 4 Subtracting does not give the scale. Divide each component of AC by the matching component of AB.
Slip: (6, 12) The question asks for k, not the vector itself.
Example 3
Point C has position 3a - b. Point D has position a - b. Write the vector CD.
3 marks
CD = OD - OC
(a - b) - (3a - b) = a - b - 3a + b
-2a
Answer: -2a
Common slips
Slip: CD = OC + OD From C to D, subtract the start.
Slip: 3a - b - a + b That is OC - OD, the journey from D to C.
Slip: 2a The b terms cancel and a - 3a = -2a.
Questions and answers
GCSE Maths vector proof questions questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Position vectors OA = a and OB = b. Write OM for the midpoint M of AB.
Show answer
Answer: 1/2(a+b)
Halve the sum of the position vectors.
Do not stop at a + b.
M1 — Halve the sum of the position vectors.
A1 — Answer 1/2(a+b)
Question 2 Easy2 marks
OA = a and OB = b. Write the journey from A to B.
Show answer
Answer: b-a
Subtract the start from the end.
b - a.
M1 — Subtract the start from the end.
A1 — Answer b-a
Question 3 Medium2 marks
AB = (2, 5) and AC = (6, 15). Are they parallel? If so, give the scalar from AB to AC.
Show answer
Answer: 3
15 / 5 = 3 and 6 / 2 = 3.
The scalar is 3, so they are parallel.
M1 — 15 / 5 = 3 and 6 / 2 = 3.
A1 — Answer 3
Question 4 Medium2 marks
Simplify the position 2a + 2b by taking out a common factor. Write the factorised form.
Show answer
Answer: 2(a+b)
2 is a factor of both terms.
The factorised form is 2(a+b).
M1 — 2 is a factor of both terms.
A1 — Answer 2(a+b)
Question 5 Hard2 marks
C is the point with position 3a - b and A is the point with position a. Write vector AC.
Show answer
Answer: 2a-b
AC = OC - OA.
(3a - b) - a = 2a - b.
M1 — AC = OC - OA.
A1 — Answer 2a-b
Question 6 Hard2 marks
CD is parallel to a and points the opposite way, with twice the length. Write CD in terms of a.
Show answer
Answer: -2a
The scalar is -2.
CD = -2a.
M1 — The scalar is -2.
A1 — Answer -2a
Question 7 Exam-style5 marks
AB = (2, 4) and AC = (6, 12). Show that C lies on the line AB by finding the scalar, and state the scalar.
Show answer
Answer: 3
6 = 3 times 2.
12 = 3 times 4.
The scalar is 3.
They share A, so C is on AB.
Parallel and a common point means collinear with A.
M1 — Compare first components
M1 — Compare second components
A1 — Scalar 3
M1 — Shared point A
A1 — Collinear
Common mistakes
Most lost marks on this topic are one slip.
Writing the midpoint as a + b with no half
The midpoint is the average of the two position vectors, so halve the sum
Finding AB as a - b
AB is the end minus the start, so AB = b - a
Claiming parallel from one matching component
Check that every component is multiplied by the same scalar
Spot the slip
M is the midpoint of AB. OA is a and OB is b. A student writes OM = a + b. What is missing?
The half
The letters should be numbers
Midpoints use subtraction only
Show the slip
The half The midpoint is the average, so half of a + b.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write the route as letters first (MX=MO+OX) before substituting — it earns a method mark and prevents sign errors.
For a collinear proof the last line must contain three ideas: a multiple, 'parallel', and 'common point'. Write all three.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.