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GCSE Maths / Edexcel

Vector ratios and proof

Use vector ratios and scalar multiples to prove points are collinear or lines are parallel.

Geometry and MeasuresHigherGrades 7 to 9Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Vector proof

Pearson Edexcel GCSE Maths Higher geometry G25: apply vector arithmetic and vector proof in geometric problems.

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Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

Vector proof is not about drawing accurately. It is about writing a route using vector notation.

If a point P divides AB in the ratio 2:3, then AP is 2/5 of AB and PB is 3/5 of AB.

A position vector starts from the origin. If OA = a and OB = b, then AB = b - a.

To prove two lines are parallel, show that one vector is a scalar multiple of the other.

To prove points are collinear, show that the vectors along the same line are scalar multiples and share a point.

Higher note: keep vector letters bold or clearly labelled in your working. The route matters more than the final-looking expression.

Edexcel proof questions often reward a clear conclusion sentence, such as therefore AB is parallel to CD.

Key ruleAB = position vector of B minus position vector of A. Parallel proof needs one vector as a scalar multiple of another.

Diagram guide

APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
Ratio on a lineIf AP:PB = 2:3, the whole line is split into 5 equal parts.
ABCDpkpshow one vector is a multiple of anotherthen the lines are parallel
Parallel proofShow that two direction vectors are scalar multiples.

Worked examples

Find a ratio vector

P lies on AB so that AP:PB = 2:3. Write AP as a fraction of AB.

APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
Example: ratio on a lineThe whole line has 2 + 3 = 5 parts.
  1. Total parts = 2 + 3 = 5.
  2. AP is 2 of the 5 parts.
  3. AP = 2/5 AB.

Answer: AP = 2/5 AB

Use position vectors

OA = a and OB = b. P divides AB in the ratio 2:3. Find OP.

APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
Example: position vector of PMove from O to A, then 2/5 of the way from A to B.
  1. AB = b - a.
  2. AP = 2/5(b - a).
  3. OP = OA + AP.
  4. OP = a + 2/5(b - a) = 3/5a + 2/5b.

Answer: OP = 3/5a + 2/5b

Prove lines are parallel

Vector AB = 3p - 6q and vector CD = p - 2q. Prove AB is parallel to CD.

ABCDpkpshow one vector is a multiple of anotherthen the lines are parallel
Example: scalar multipleFactor the larger vector.
  1. AB = 3p - 6q.
  2. AB = 3(p - 2q).
  3. CD = p - 2q.
  4. So AB = 3CD, therefore AB is parallel to CD.

Answer: AB is parallel to CD because AB = 3CD.

Common mistakes

  • Using 2/3 instead of 2/5 for a 2:3 ratio.
  • Writing AB = a - b instead of b - a.
  • Showing two vectors have similar terms but not proving one is a scalar multiple.
  • Forgetting the conclusion sentence in a proof question.
  • Treating a position vector like a side vector without checking its start point.

Quick exercise

Try these before moving to the exam-style questions.

  1. P divides AB in the ratio 1:4. What fraction of AB is AP?
    APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
    Quick check: ratio fractionUse the total number of parts.
  2. If OA = a and OB = b, write AB.
    ABCDpkpshow one vector is a multiple of anotherthen the lines are parallel
    Quick check: position vectorsFinish minus start.
  3. Are vectors 4p + 8q and p + 2q parallel?
    ABCDpkpshow one vector is a multiple of anotherthen the lines are parallel
    Quick check: scalar multipleLook for a common multiplier.
  4. What must you show to prove two lines are parallel using vectors?
    ABCDpkpshow one vector is a multiple of anotherthen the lines are parallel
    Quick check: proof conditionParallel direction vectors are scalar multiples.
  5. Q divides AB in the ratio 3:2. What fraction of AB is AQ?
    APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
    Quick check: line ratioThe whole line has 5 parts.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleHigherNon-calculator3 marks

OA = a and OB = b. Point P lies on AB so that AP:PB = 1:3. Find OP in terms of a and b.

APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
Question diagram: ratio pointMove from A one quarter of the way to B.
vector ratiosposition vectorshigher proof
Standard exam styleHigherNon-calculator4 marks

Vector AB = 2p + 6q and vector CD = p + 3q. Prove that AB is parallel to CD.

ABCDpkpshow one vector is a multiple of anotherthen the lines are parallel
Question diagram: parallel vectorsFactor one vector to show a scalar multiple.
vector proofparallel linesscalar multiple
ChallengeHigherNon-calculator5 marks

OA = a and OB = b. P divides AB in the ratio 2:1. Show that OP = 1/3a + 2/3b.

APB2 parts3 partsAP:PB = 2:3 means AP is 2/5 of ABposition vector of P uses the same fraction
Question diagram: show thatUse the ratio to move 2/3 of the way from A to B.
vector proofratio on a lineshow that