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4024/11 May/Jun 2022, Question 25: Vector Proofs

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 6 marks

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The question, in short

In triangle ACD, B and E are midpoints of AC and AD, with vectors AB = 6a + 3b and DC = 5a + 2b. Find AC and AD, then show EB is parallel to DC.

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Part (ai) · 1 mark

Find vector AC in terms of a and b, using that B is the midpoint of AC.

  1. 1AC = 2 × AB = 2(6a + 3b)
  2. 2= 12a + 6b

Answer12a + 6b or 6(2a + b)

Part (aii) · 2 marks

Find vector AD in terms of a and b, using midpoint E and the given DC.

  1. 1B is the midpoint of AC, so AC = 2 × AB = 12a + 6b.
  2. 2AD = AC + CD, and CD = −DC = −5a − 2b.
  3. 3AD = 12a + 6b − 5a − 2b = 7a + 4b

Answer7a + 4b

Part (b) · 3 marks

Show that vector EB is parallel to vector DC.

  1. 1AD = AC + CD = (12a + 6b) − (5a + 2b) = 7a + 4b
  2. 2E is the midpoint of AD, so EA = −(1/2)AD = −3.5a − 2b
  3. 3EB = EA + AB = −3.5a − 2b + 6a + 3b = 2.5a + b
  4. 42.5a + b = (1/2)(5a + 2b), so EB = (1/2)DC
  5. 5→EB = 2.5a + b, a multiple of →DC, so EB is parallel to DC

Answer→EB = 2.5a + b →EB is parallel to →DC because →EB = k→DC

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Also searched as: 4024/11/M/J/22 · 4024_s22_qp_11 · 4024 s22 11 · question 25.
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