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0580/42 Oct/Nov 2021, Question 9: Magnitude of a Vector

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 9 marks

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The question, in short

Vector geometry: use a given point and column vector to find coordinates, a scaled vector and a length, then work with a parallelogram defined by vectors a and c.

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

Part (ai) · 1 mark

Point F is (5, −2) and vector FG is (−2, 3). Find the coordinates of G.

  1. 1G = (5 − 2, −2 + 3) = (3, 1)

Answer(3, 1)

Part (aii) · 1 mark

Find the vector 5 times FG, where FG is the column vector (−2, 3).

  1. 15 × (−2, 3) = column vector (−10, 15)

Answercolumn vector (−10, 15)

Part (aiii) · 2 marks

Find the magnitude of vector FG, which is the column vector (−2, 3).

  1. 1|FG| = √((−2)² + 3²) = √13
  2. 2= 3.605… ≈ 3.61

Answer3.61 or 3.605 to 3.606

Part (bia) · 1 mark

In parallelogram OABC with OA = a and OC = c, Q is the midpoint of AB. Find vector AQ in terms of a and/or c.

  1. 1AQ = 1/2 AB = 1/2 c

Answer1/2 c

Part (bib) · 1 mark

Find vector OQ in terms of a and/or c, where Q is the midpoint of AB.

  1. 1OQ = OA + AQ = a + 1/2 c

Answera + 1/2 c

Part (biia) · 2 marks

Given OP = 2/3 a + 1/3 c, show that points O, P and Q are collinear.

  1. 1OP = 2/3 a + 1/3 c = 1/3(2a + c)
  2. 2OQ = a + 1/2 c = 1/2(2a + c)
  3. 3So OQ = 1.5 × OP: OP and OQ are parallel and share the point O, so O, P and Q lie on a straight line

Answer→OP = 1/3(2a + c) and →OQ = 1/2(2a + c) OR →OP = 2/3(a + 1/2c) OR →PQ = 1/3(a + 1/2c), and correct comment e.g. have the same base vector or that they are multiples of one another and they share a common point OR e.g. →OQ = 1.5→OP, 2→PQ = →OP

Part (biib) · 1 mark

Write the ratio OP : OQ in the form 1 : n.

  1. 1OP : OQ = 1/3 : 1/2 = 1 : 1.5

Answer1 : 1.5

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