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
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 12 of 12 in this topic. Use this topic summary 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 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.