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0580/22 May/Jun 2026, Question 22: Expressing Vectors in Terms of a and b

Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Non-calculator · 5 marks

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The question, in short

In parallelogram OABC, OA = a and OC = c, with P on diagonal AC dividing AP : PC = 1 : 4. Express vectors in terms of a and c, then prove O, P, Q are collinear.

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

Part (ai) · 2 marks

Write vector AP using a and c, given that AP is one fifth of AC.

  1. 1→AC = −a + c = c − a
  2. 2AP is 1/5 of AC
  3. 3→AP = 1/5(c − a)

Answer1/5(c − a) or 1/5 c − 1/5 a

Part (aii) · 1 mark

Write vector OP in terms of a and c in simplest form.

  1. 1→OP = →OA + →AP = a + 1/5 c − 1/5 a
  2. 2= 4/5 a + 1/5 c

Answer1/5(4a + c) or 4/5 a + 1/5 c

Part (b) · 2 marks

Given Q on AB with AQ = 1/4 c, demonstrate that O, P and Q are collinear.

  1. 1→OQ = →OA + →AQ = a + 1/4 c = 1/4(4a + c)
  2. 2→OP = 1/5(4a + c)
  3. 3So →OQ = 5/4 →OP: they are parallel
  4. 4They share the point O, so O, P and Q lie in a straight line

Answer→OP = 1/5(4a + c) and →OQ = 1/4(4a + c), so →OQ = 5/4 →OP; →OQ is parallel to →OP and they share point O, so O, P and Q are collinear

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Also searched as: 0580/22/M/J/26 · 0580_s26_qp_22 · 0580 s26 22 · question 22.
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