This topic summary covers Knowledge Organiser: Vectors — Geometry Proofs 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 2 of 12 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Vectors — Geometry Proofs
Key Terms
- Collinear: Three or more points lying on the same straight line
- Parallel vectors: One vector is a scalar multiple of the other
- Position vector: A vector from the origin O to a point
- Midpoint: OM = ½(OA + OB) — average of position vectors
- Ratio division: P divides AB in ratio m:n → OP = OA + [m/(m+n)] × AB
Must-Know Facts
- To find any vector, follow a PATH through labelled points
- Travelling against an arrow: negate the vector
- Parallel lines: show one vector = k × the other, state k
- Collinear points: show vectors are parallel AND share a common point
- Always write a conclusion sentence for proof questions
Key Methods
- Path rule: AB = AO + OB = −OA + OB = −a + b
- Midpoint: OM = ½(a + b)
- Parallel test: p = kq for non-zero scalar k
- Collinear: parallel + common point (state both)
Key Formulas
- AB = b − a (position of end minus position of start)
- Midpoint M of AB: OM = ½(a + b)
- Parallel vectors: p = kq for some non-zero scalar k
- Collinear points: show vectors are parallel AND share a common point
Common Mistakes
- AB = a − b: Wrong — AB = b − a (end position minus start position)
- Parallel but not collinear: Parallel vectors may not lie on the same line — must also show a common point for collinearity
- Not showing working clearly: Write each vector path step by step using the rules — examiners award marks for method
- Scalar multiple vs sum: For parallel proof, express one vector as a scalar multiple of the other — not just show they have the same direction
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.