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0580/21 May/Jun 2026, Question 25: Vector Geometry

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 7 marks

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The question, in short

Triangle ABC has line ACD extended; P on AB with AP:PB = 4:1, Q on BC with BQ = 2/5 BC, AB = m, AC = 5n. Express vectors and prove collinearity.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (ai) · 1 mark

Using m and n, express vector BC in simplest form.

  1. 1→BC = →BA + →AC = −m + 5n
  2. 2= 5n − m

Answer5n − m

Part (aii) · 3 marks

Using m and n, express vector PQ in simplest form.

  1. 1→PB = 1/5 m
  2. 2→BQ = 2/5(5n − m) = 2n − 2/5 m
  3. 3→PQ = 1/5 m + 2n − 2/5 m
  4. 4= 2n − 1/5 m

Answer2n − 1/5 m or 1/5(10n − m)

Part (b) · 3 marks

Given CD = 3n, show that points P, Q and D are collinear.

  1. 1→QC = 3/5(5n − m) = 3n − 3/5 m
  2. 2→QD = →QC + →CD = 3n − 3/5 m + 3n = 6n − 3/5 m
  3. 3→QD = 3(2n − 1/5 m) = 3→PQ
  4. 4→QD is parallel to →PQ and they share the point Q
  5. 5So P, Q and D lie on a straight line

Answer→QD = 6n − 3/5 m = 3(2n − 1/5 m) = 3 →PQ, so →QD is parallel to →PQ and they share point Q, so P, Q and D are collinear

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