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Proving Two Lines Are Parallel

Part of Vectors (Geometry Proofs) · GCSE GCSE Mathematics revision

This deep dive covers Proving Two Lines Are Parallel within Vectors (Geometry Proofs) for GCSE Mathematics. Revise Vectors (Geometry Proofs) in Geometry & Measures for GCSE Mathematics with 14 exam-style questions and 12 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 7 of 12 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.

Topic position

Section 7 of 12

Practice

14 questions

Recall

12 flashcards

Proving Two Lines Are Parallel

Two vectors are parallel if one is a scalar multiple of the other. That is, vector p is parallel to vector q if p = kq for some non-zero scalar k.

Method for a proof question:

  1. Express both direction vectors in terms of a and b by following paths through the diagram.
  2. Simplify both vectors.
  3. Show that one is a scalar multiple of the other and state what the scalar is.
  4. Write a concluding sentence: "Since PQ = k×RS, PQ is parallel to RS."

Worked Example: In triangle OAB, OA = a, OB = b. P is the midpoint of OA. Q is the midpoint of OB. Prove PQ is parallel to AB.

PQ = PO + OQ = -½a + ½b = ½(-a + b) = ½AB

Since PQ = ½AB, PQ is parallel to AB (and half its length).

Keep building this topic

Read this section alongside the surrounding pages in Vectors (Geometry Proofs). That gives you the full topic sequence instead of a single isolated revision point.

Worked Example: Midpoint VectorProving Three Points Are CollinearVectors in Geometry: Proving What You Can SeeKnowledge Organiser: Vectors — Geometry Proofs

Practice Questions for Vectors (Geometry Proofs)

Vector AB goes from point A to point B. Which of the following describes vector BA?

  • A. The same vector as AB
  • B. Twice the length of AB in the same direction
  • C. The same magnitude as AB but in the opposite direction
  • D. Half the length of AB in the opposite direction
1 markfoundation

A student says: 'I have shown that vector AB is parallel to vector CD, so A, B, C, D all lie on the same straight line.' Explain why the student's reasoning is incorrect.

2 marksstandard

Quick Recall Flashcards

If AB = a, what is BA?
BA = -a. Reversing the direction of travel negates the vector. In diagrams: if you travel against an arrow, write a negative sign in front of that vector.
When are two vectors parallel?
Two vectors p and q are parallel when one is a scalar multiple of the other: p = kq for some non-zero scalar k. They point in the same or exactly opposite direction.

14 questions on Vectors (Geometry Proofs) — practise free

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Worked Example: Midpoint Vector

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Proving Three Points Are Collinear