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.
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:
- Express both direction vectors in terms of a and b by following paths through the diagram.
- Simplify both vectors.
- Show that one is a scalar multiple of the other and state what the scalar is.
- 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).
Practice questions for Vectors (Geometry Proofs)
Vector AB goes from point A to point B. Which of the following describes vector BA?
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.