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4PM1/01R Jan 2022, Question 6: Vector Proofs

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 10 marks

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The question, in short

Given vectors PQ = 2i + 3j, PR = −i + 18j and PS = −3i + 15j for quadrilateral PQRS, prove it is a parallelogram, find a unit vector along QS, and find PT with T on QS in ratio QT : TS = 5 : 8.

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Part (a) · 4 marks

Use the three given vectors to demonstrate that the quadrilateral PQRS satisfies the conditions for being a parallelogram.

  1. 1SR = SP + PR = (3i − 15j) + (−i + 18j) = 2i + 3j
  2. 2So SR = PQ
  3. 3QR = QP + PR = (−2i − 3j) + (−i + 18j) = −3i + 15j = PS
  4. 4Opposite sides are equal and parallel, so PQRS is a parallelogram

AnswerSR = PQ = 2i + 3j (and QR = PS = −3i + 15j), so PQRS is a parallelogram

Part (b) · 4 marks

Determine a unit vector pointing in the same direction as QS, giving it in simplified i, j form.

  1. 1QS = QP + PS = (−2i − 3j) + (−3i + 15j) = −5i + 12j
  2. 2|QS| = √(25 + 144) = 13
  3. 3Unit vector = (1/13)(−5i + 12j)

Answer(1/13)(−5i + 12j)

Part (c) · 2 marks

Given T divides QS with QT : TS = 5 : 8, express the vector PT in simplified i, j form.

  1. 1PT = PQ + (5/13)QS
  2. 2= 2i + 3j + (5/13)(−5i + 12j)
  3. 3= (1/13)i + (99/13)j

AnswerPT = (1/13)i + (99/13)j

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