Three points A, B, C are collinear (lie on the same straight line) if:

Vector AB is parallel to vector BC (a scalar multiple), AND
They share a common point (point B).

Both conditions are needed. Parallel alone is not enough — the vectors could be on separate parallel lines.

Method:

Find AB and BC (or AC) in terms of a and b.
Show AB = k × BC for some scalar k.
State that since AB and BC are parallel and share point B, A, B, C are collinear.

This deep dive covers Proving Three Points Are Collinear 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 8 of 12 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.

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

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.

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.

Proving Three Points Are Collinear

Practice questions for Vectors (Geometry Proofs)

Quick recall flashcards

14 questions on Vectors (Geometry Proofs): practise free