Home   >  Worksheets   >  GCSE questions by topic   >  Vector proof

MathsLinks Logo Questions by topic

Last update: 05/11/2026

Give feedback : Buy me a coffee

Vector proof

Free Printable Vector proof PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 8+)

Download free vector proof worksheets (PDF), GCSE maths past paper questions and worked solutions for Vector proof (8K). This is a Grade 8+ geometry topic from the GCSE questions by topic collection for Foundation and Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Maths Made Easy, Dr Frost, Naiker Maths, 1st Class Maths, Maths4EveryOne, Metatutor, Maths Takeaway, Dr Austin, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our vector proof revision notes and worked examples with diagrams, try real GCSE past paper vector proof questions with step-by-step solutions, or download a PDF of all vector proof worksheet links.

Free Vector proof PDF Worksheets & Answers

Grade 8+ • Topic 8K
Resource Questions Answers
GCSE past paper questions & solutions
 Qs Ans
Graded exam questions & solutions
 Qs Ans
MME worksheet & model answers
 Qs Ans
GCSE worksheet & solutions
 Qs Ans
Exam question booklet & solutions
 Qs Ans
Edexcel exam questions & solutions
 Qs Ans
AQA exam questions & solutions
 Qs Ans
GCSE 9–1 practice booklet & solutions
 Qs Ans
Topic worksheet & worked solutions
 Qs Ans
GCSE Maths Takeaway exam questions
 Qs Ans
Practice grid / worksheet & answers
 Qs Ans
Practice questions & answers
 Qs Ans
GCSE topic booklet & answers
 Qs Ans
PMT (Edxl)
Edexcel GCSE topic questions & solutions
 Qs Ans
PMT (AQA)
AQA GCSE topic questions & solutions
 Qs Ans
PMT (IGCSE)
Edexcel IGCSE topic questions & solutions
 Qs Ans
Interactive online lesson & practice
 Page
Grade 8+ topics
8A - Quad. sim. equations8B - Transforming graphs y=f(x)
8C - Algebraic proof8D - Completing the square
8E - Quadratic inequalities8F - Quadratic sequences
8G - Velocity time graphs8H - Circles and tangents
8J - Circle theorem proof8K - Vector proof
8L - Probability equation questions8M - Invariant points
8N - Spheres, cones and cylinders8P - Volume of a frustum
GCSE Maths Grade 8+ Revision Notes & Method

Guide to Vector proof: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To find a vector between two points, trace a known path along the diagram, reversing the sign of a vector when travelling against its arrow (e.g. AB = AO + OB = −a + b = b − a). If a point divides a line in the ratio m : n, use the fraction mm + n of the full vector.

To prove two vectors are parallel, factorise both so they are scalar multiples of the same bracketed vector (e.g. PQ = k × RS). If they also share a common point (such as P, Q, R where PQ = k × QR), then the three points are collinear (lie on a straight line).

Example 1: In ΔOAB, OA = a and OB = b. M is the midpoint of OA and N divides OB in the ratio 1 : 2. Find MN in terms of a and b.

O A B M N a b

Solution:

MO = −12a;  ON = 13b (ratio 1 : 2)
MN = MO + ON ⇒ Answer: 13b − 12a

Example 2: Given PQ = 2a + 4b and QR = 3a + 6b, prove that P, Q, and R lie on a straight line.

Solution:

Factorise both vectors:
PQ = 2(a + 2b) and QR = 3(a + 2b)
Therefore QR = 32PQ, so PQ and QR are parallel
Since they also share the common point Q, P, Q, and R are collinear.
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Vector proof Questions & Answers

Past Paper Example 1 AQA Nov 2022 2H Q25 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
Two congruent parallelograms, PQRV and VRST, are joined.
abPVXTQRSWNot drawnaccurately
QP→ = a PV→ = b
X is the midpoint of VT.
VW : WR = 1 : 2
Prove that Q, W and X lie on a straight line.
Worked Solution & Mark Scheme (3 marks)
VR = PQ = –a M1
QW = QP + PV + VW = a + b – 1/3 a = 2/3 a + b
QX = QP + PV + VX = a + b + 1/2 b = a + 3/2 b M1
QX = 3/2 (2/3 a + b) = 3/2 QW
Since QX is a multiple of QW and Q is a common point, Q, W and X are collinear. C1
Answer: QW = a + b – 1/3 a = 2/3 a + b.
QX = a + 3/2 b = 3/2 (2/3 a + b) = 3/2 QW.
Since QX is a scalar multiple of QW and they share point Q, Q, W and X lie on a straight line.
Past Paper Example 2 AQA Jun 2024 2H Q23 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
DEFC6a + b2a – 5b4a – 6bNot drawnaccurately
Vector diagram with DC = 6a + b, CE = 2a – 5b, CF = 4a – 6b. Prove that DEF is a straight line. [4 marks]
Prove that DEF is a straight line.
Worked Solution & Mark Scheme (4 marks)
DE = DC + CE = (6a + b) + (2a – 5b) = 8a – 4b = 4(2a – b) M2
EF = EC + CF = –(2a – 5b) + (4a – 6b) = 2a – b M1
Since DE = 4EF, DE and EF are parallel and share common point E, so DEF is a straight line. C1
Answer: DE = 8a – 4b = 4(2a – b) and EF = 2a – b, so DE = 4EF; parallel and share point E, so DEF is a straight line
Past Paper Example 3 AQA Nov 2023 1H Q28 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
PQRS is a quadrilateral.
PQ is not parallel to SR.
X is a point on QR.
QX : XR = 2 : 3
QX→ = 2a + 4b
PSQXR5a2a + 4ba – 8bNot drawnaccurately
Prove that PQRS is a trapezium.
Worked Solution & Mark Scheme (3 marks)
Since QX : XR = 2 : 3, QR = (5/2)QX = (5/2)(2a + 4b) = 5a + 10b M1
PS = PQ + QR + RS = –5a + (5a + 10b) + (a – 8b) = a + 2b M1
Since QR = 5(a + 2b) = 5PS, QR is parallel to PS, so PQRS has a pair of parallel sides and is a trapezium. C1
Answer: QR = 5/2(2a + 4b) = 5a + 10b
PS = –5a + (5a + 10b) + (a – 8b) = a + 2b
QR = 5PS so QR is parallel to PS, hence PQRS is a trapezium

← 8J Circle theorem proof

Back to GCSE questions by topic

8L Probability equation questions →