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0580/42 May/Jun 2021, Question 5: Expressing Vectors in Terms of a and b

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 16 marks

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The question, in short

Vector question: given column vectors a and b, compute combinations and a modulus, then use a vector equation to find k and m; in a parallelogram with p and q, find vectors and a meeting point.

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

Part (aia) · 1 mark

Find b − a as a column vector.

  1. 1b − a = (2 − (−3), −5 − 8) = (5, −13)

Answercolumn vector (5, −13)

Part (aib) · 2 marks

Find 2a + b as a column vector.

  1. 12a = (−6, 16)
  2. 22a + b = (−6 + 2, 16 − 5) = (−4, 11)

Answercolumn vector (−4, 11)

Part (aic) · 2 marks

Find the modulus |b|.

  1. 1|b| = √(2² + (−5)²) = √29
  2. 2= 5.385… ≈ 5.39

Answer5.39 or 5.385…

Part (aii) · 3 marks

Given a + kb = (13, m), find integers k and m.

  1. 1x: −3 + 2k = 13, so k = 8
  2. 2y: m = 8 − 5 × 8 = −32
  3. 3k = 8, m = −32

Answerk = 8, m = −32

Part (bia) · 1 mark

Express vector OB in terms of p and q.

  1. 1→OB = →OA + →AB = p + q

Answerp + q

Part (bib) · 2 marks

Express vector CM in terms of p and q, simplified.

  1. 1→CM = →CO + →OM = −q + 1/2(p + q)
  2. 2= 1/2 p − 1/2 q

Answer1/2p − 1/2q or 1/2(p − q) or (p − q)/2

Part (bic) · 2 marks

Express vector MN in terms of p and q, simplified.

  1. 1→BN = −2/5 q
  2. 2→MN = →MB + →BN = 1/2(p + q) − 2/5 q
  3. 3= 1/2 p + 1/10 q

Answer1/2p + 1/10q or (5p + q)/10

Part (bii) · 3 marks

Lines CB and ON extended meet at D; find D's position vector in terms of p and q.

  1. 1→ON = p + 3/5 q, so D = λ(p + 3/5 q)
  2. 2D is on CB, where the q-part is q: 3/5 λ = 1, λ = 5/3
  3. 3Position vector of D = 5/3 p + q

Answer5/3p + q or (5p + 3q)/3

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Also searched as: 0580/42/M/J/21 · 0580_s21_qp_42 · 0580 s21 42 · question 5.
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