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4PM1/01R Jun 2019, Question 7: Vector Proofs

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

Given position vectors OA = 8i − 6j, OB = 15i − 6j and OC = 8i + j for fixed points, work out a vector, a unit vector, and prove three points lie on one line.

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

Part (a) · 2 marks

Work out the vector from B to C, giving it in i and j form in simplest terms.

  1. 1BC = BO + OC
  2. 2= −(15i − 6j) + 8i + j = −7i + 7j

AnswerBC = −7i + 7j

Part (b) · 2 marks

Work out a unit vector pointing the same way as BC.

  1. 1|BC| = √(49 + 49) = √98 = 7√2
  2. 2Unit vector = (1/(7√2))(−7i + 7j) = (1/√2)(−i + j)

Answer(1/√98)(−7i + 7j)

Part (c) · 4 marks

M is halfway along OA and N splits OB in ratio 1 : 2; show M, N and C lie on a single straight line.

  1. 1OM = 4i − 3j, ON = (1/3)OB = 5i − 2j
  2. 2MN = (5i − 2j) − (4i − 3j) = i + j
  3. 3MC = (8i + j) − (4i − 3j) = 4i + 4j = 4MN
  4. 4MN and MC are parallel and share the point M, so M, N and C are collinear

AnswerMN = i + j and MC = 4i + 4j = 4MN, so M, N, C are collinear

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