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4024/21 Oct/Nov 2020, Question 9: Expressing Vectors in Terms of a and b

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 12 marks

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The question, in short

Given points H(5, 2) and J(−3, 6), work with the vector from H to J, and use a seven-triangle diagram with OA = p and OF = q to express vectors.

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

Part (ai) · 1 mark

Work out the vector going from H to J.

  1. 1→HJ = (−3 − 5, 6 − 2) = column vector (−8, 4)

Answercolumn vector (−8, 4)

Part (aii) · 2 marks

Work out the length of the vector from H to J.

  1. 1→HJ = (−3 − 5, 6 − 2) = (−8, 4)
  2. 2|→HJ| = √((−8)² + 4²) = √(64 + 16) = √80
  3. 3√80 = 8.944...

Answer8.944...

Part (aiii) · 2 marks

Given M is halfway between H and J, work out M's position vector.

  1. 1M = ((5 + (−3))/2, (2 + 6)/2) = (1, 4)
  2. 2Position vector of M = column vector (1, 4)

Answercolumn vector (1, 4)

Part (bia) · 1 mark

Using p and/or q, write the vector from F to B in simplest form.

  1. 1→FG = p and →GB = p
  2. 2→FB = 2p

Answer2p

Part (bib) · 1 mark

Using p and/or q, write the vector from F to E in simplest form.

  1. 1→FE = →OG = →OA + →AG = p + q

Answerp + q

Part (bii) · 2 marks

Given X lies on FB with FX : XB = 3 : 1, write the vector from O to X using p and/or q.

  1. 1→FX = 3/4 × 2p = 3/2 p
  2. 2→OX = →OF + →FX = q + 3/2 p = 3/2 p + q

Answer3/2 p + q

Part (biii) · 3 marks

Given OXYF is a trapezium with X on FB (FX : XB = 3 : 1) and Y on BD, write the vector from X to Y using p and/or q.

  1. 1OXYF is a trapezium with FY parallel to OX, and Y is on BD so →BY = kq
  2. 2→FY = 2p + kq must be a multiple of 3/2 p + q: k = 4/3, →FY = 2p + 4/3 q
  3. 3→XY = →FY − →FX = 2p + 4/3 q − 3/2 p
  4. 4= 1/2 p + 4/3 q

Answer1/2 p + 4/3 q

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