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4024/12 May/Jun 2025, Question 21: Expressing Vectors in Terms of a and b

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 5 marks

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The question, in short

In triangle OAB with vectors OA = a and OB = b, X divides AB in ratio 3:2, OXC is straight, and AC is parallel to OB. Find AX and XC in terms of a and b.

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

Part (a) · 2 marks

Find vector AX in its simplest form using a and b, given AX:XB = 3:2.

  1. 1→AB = b − a
  2. 2→AX = 3/5 →AB = 3/5(b − a)

Answer3/5(b − a) or 3/5 b − 3/5 a

Part (b) · 3 marks

Find vector XC in its simplest form using a and b, using the straight line OXC and AC parallel to OB.

  1. 1→OX = a + 3/5(b − a) = 2/5 a + 3/5 b
  2. 2AC is parallel to OB, so →OC = a + kb; OC is a multiple of OX, so →OC = 5/2 →OX = a + 3/2 b
  3. 3→XC = →OC − →OX = (a + 3/2 b) − (2/5 a + 3/5 b)
  4. 4= 3/5 a + 9/10 b

Answer3/5 a + 9/10 b

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Also searched as: 4024/12/M/J/25 · 4024_s25_qp_12 · 4024 s25 12 · question 21.
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