Sheet № 217 · Higher only · AQA · Edexcel · OCR
Vectors: Proving Parallel and Collinear –
Proving that lines are parallel or that points are collinear using vectors is a challenging Higher-tier topic tested on AQA, Edexcel, and OCR GCSE Maths papers. These proof questions require you to find vector expressions for line segments and then compare them to show a relationship. This guide explains the key principles, walks through
§Key definitions
Question:
OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.
Answer:
OM = ½a + ½b.
Q1 (Higher):
OA = a and OB = b. M is the midpoint of OA and N is the midpoint of OB. Prove that MN is parallel to AB.
Q2 (Higher):
OA = 6a and OB = 6b. P divides OA in the ratio 1:2 and Q divides OB in the ratio 1:2. Find PQ and show it is parallel to AB.
Q3 (Higher):
OA = a and OB = b. X lies on AB such that AX:XB = 1:3. Express OX in terms of a and b.
§Formulas to memorise
If AB = k × CD, then AB is parallel to CD
If AB = k × AC and both share point A, then A, B, and C are collinear
State your conclusion: "Since XY = k × PQ, the lines XY and PQ are parallel."
Show that AC = k × AB for some scalar k.
State: "Since AC = k × AB and both start from A, the points A, B, and C are collinear."
Worked example
OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.
This topic is Higher only, but this example uses a basic setup.
⚠ Common mistakes
- ✗Not simplifying fully. You must factorise your vector expression completely to reveal the scalar multiple. If you leave it unsimplified, the relationship is hidden.
- ✗Forgetting to state the conclusion. You must explicitly write "therefore the lines are parallel" or "therefore the points are collinear" — the scalar multiple alone does not earn the final mark.
- ✗Getting direction wrong. AB = b − a (not a − b) when OA = a and OB = b. Think of it as "destination minus start."
✦ Exam tips
- →Always define your route clearly: state which vectors you are adding (e.g. PQ = PO + OQ).
- →Factorise your final vector expression — the scalar factor proves parallelism.
- →For collinear proofs, you need two things: (1) the vectors are parallel (scalar multiple), and (2) they share a common point.
- →Draw a clear diagram and label all given vectors — this helps you plan your route.