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Vectors in Geometry: Proving What You Can See
Vectors in Geometry: Proving What You Can See
In a GCSE exam, a vector diagram might show a triangle or quadrilateral with some sides labelled a and b. The question asks you to prove that two points lie on the same straight line, or that one line is parallel to another. You cannot just say "it looks parallel" — you need algebra. Vector geometry gives you the tools to do exactly that. These questions appear on Higher tier papers and are worth 4-6 marks.
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.
Mathematics glossary
- 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.
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Exam Tips
- Always show the route: write AB = AO + OB before substituting. Examiners want to see the path, not just the answer.
- Factor out clearly: write ½(-a + b) rather than leaving it unexpanded — it makes the scalar multiple obvious.
- State the conclusion: never let the algebra speak for itself. Write "Therefore PQ is parallel to AB" or "Therefore A, B, C are collinear."
- Watch the direction of arrows: if the arrow in the diagram points from B to A but you need AB, that is -BA.
- Collinearity needs two conditions: parallel direction AND a shared point. Say both explicitly.
- Common error: writing OA = a when OA means the magnitude. Context matters — in vector geometry OA usually means the vector.
Now try it yourself
Quiz · Question 1 of 14
Vector AB goes from point A to point B. Which of the following describes vector BA?
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This topic in real past papers
Every real exam question we've found on vectors (geometry proofs), with a full worked answer.
AQA Paper 1
Two of the four sittings we have full papers for give a diagram with several vectors labelled in terms of a and b, and ask for an unknown vector, constant, or straight-line proof, built from those given vectors.
Edexcel Paper 2
Two of the three sittings we have full papers for use vector notation to express one side of a shape in terms of two given vectors, and June 2019 goes further, using a straight-line condition to find an unknown ratio.