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0580/43 Oct/Nov 2022, Question 10: Vector Geometry

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 11 marks

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The question, in short

Use vectors a = (1, 2) and b = (−3, 5) to draw scaled and combined vectors, then work with a trapezium OABC to find position vectors.

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

Part (ai) · 1 mark

Draw and label the vector 2a on the grid.

  1. 12a = column vector (2, 4)
  2. 2Draw an arrow 2 right and 4 up, label it 2a

Answer2a drawn correctly with direction arrow

Part (aii) · 2 marks

Draw and label the vector a − b on the grid.

  1. 1a − b = (1 − (−3), 2 − 5) = column vector (4, −3)
  2. 2Draw an arrow 4 right and 3 down, label it a − b

Answera − b drawn correctly with direction arrow

Part (bia) · 1 mark

Find the position vector OB in terms of p and q.

  1. 1→OB = →OC + →CB = q + 3/4 p

Answerq + 3/4 p

Part (bib) · 2 marks

Find the vector AB in terms of p and q.

  1. 1→AB = →AO + →OB = −p + q + 3/4 p
  2. 2= q − 1/4 p

Answerq − 1/4 p

Part (bic) · 3 marks

Find the vector MN in terms of p and q.

  1. 1→MB = 3/8 p, →BN = −2/3 →AB = −2/3(q − 1/4 p)
  2. 2→MN = 3/8 p − 2/3 q + 1/6 p
  3. 3= 13/24 p − 2/3 q

Answer13/24 p − 2/3 q

Part (bii) · 2 marks

Find the position vector of G where OA and MN extended meet, in terms of p.

  1. 1→OG = →OM + t→MN = q + 3/8 p + t(13/24 p − 2/3 q)
  2. 2q terms cancel: 1 − 2/3 t = 0, t = 3/2
  3. 3→OG = 3/8 p + 3/2 × 13/24 p = 3/8 p + 13/16 p
  4. 4= 19/16 p

Answer19/16 p

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Also searched as: 0580/43/O/N/22 · 0580_w22_qp_43 · 0580 w22 43 · question 10.
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