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 detailed worked examples, and provides practice questions to prepare you for the exam.

What Does It Mean for Vectors to Be Parallel or Collinear?

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other. If a = kb for some scalar k, then a and b are parallel (they have the same or exactly opposite direction).

Collinear Points

Three points A, B, and C are collinear (they lie on the same straight line) if the vector AB is parallel to the vector AC — meaning AB = k × AC for some scalar k. Since both vectors share the point A, parallelism plus a common point guarantees all three points lie on one line.

Key Principle

Step-by-Step Method

Proving Lines Are Parallel

1. Express each line segment as a vector in terms of the given base vectors (usually a and b).
2. Simplify each expression.
3. Show that one vector is a scalar multiple of the other.
4. State your conclusion: "Since XY = k × PQ, the lines XY and PQ are parallel."

Proving Three Points Are Collinear

1. Pick one of the three points as your starting point — say A.
2. Find the vector AB in terms of the base vectors.
3. Find the vector AC in terms of the base vectors.
4. Show that AC = k × AB for some scalar k.
5. State: "Since AC = k × AB and both start from A, the points A, B, and C are collinear."

Worked Example 1 — Foundation Level

This topic is Higher only, but this example uses a basic setup.

Question: OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.

Working:

Step 1 — AB = AO + OB = −a + b = b − a.

Step 2 — M is the midpoint of AB, so AM = ½AB = ½(b − a).

Step 3 — OM = OA + AM = a + ½(b − a) = a + ½b − ½a = ½a + ½b.

Answer: OM = ½a + ½b.

Worked Example 2 — Higher Level

Question: In triangle OAB, OA = a and OB = b. P is the point on OA such that OP = (2/3)a. Q is the point on OB such that OQ = (2/3)b. Prove that PQ is parallel to AB.

Working:

Step 1 — PQ = PO + OQ = −(2/3)a + (2/3)b = (2/3)(b − a).

Step 2 — AB = AO + OB = −a + b = b − a.

Step 3 — PQ = (2/3)(b − a) = (2/3) × AB.

Step 4 — Since PQ is a scalar multiple of AB, PQ is parallel to AB.

Answer: PQ = (2/3)AB, so PQ is parallel to AB.

Worked Example 3 — Exam Style

Question: OA = a and OB = b. D is the midpoint of AB. C is the point such that OC = 2a + 2b. Show that O, D, and C are collinear and find the ratio OD : DC.

Working:

Step 1 — OD = OA + AD = a + ½(AB) = a + ½(b − a) = ½a + ½b.

Step 2 — OC = 2a + 2b.

Step 3 — Compare: OC = 2a + 2b = 4(½a + ½b) = 4 × OD.

Step 4 — Since OC = 4 × OD and both vectors start from O, the points O, D, and C lie on the same straight line, so they are collinear.

Step 5 — DC = OC − OD = 4 × OD − OD = 3 × OD. Therefore OD : DC = 1 : 3.

Answer: O, D, and C are collinear with OD : DC = 1 : 3.

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.

Practice Questions

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.

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Related Topics

• Vectors — basics of vector addition, subtraction, and scalar multiplication.
• Congruence and Similarity — similar triangles and parallel sides.
• Algebraic Proof — structuring mathematical proofs.

Summary

Proving lines are parallel or points are collinear using vectors requires clear, systematic working. Express each line segment in terms of the base vectors, simplify fully, and factorise to reveal a scalar multiple. Two vectors are parallel if one equals k times the other. Three points are collinear if the vectors between them are parallel and share a common point. Always state your conclusion explicitly and show all steps — vector proof questions carry multiple marks for method.