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Exam questions · Maths · Vectors, Constructions and Loci

Column Vectors and Vector Arithmetic

  • 6 exam questions
  • 17 marks
  • 9 quick checks
  1. 1 Calculate [2 marks]

    Calculate \(\begin{pmatrix} 4 \\ -2 \end{pmatrix} - \begin{pmatrix} 1 \\ 3 \end{pmatrix}\). [2 marks]

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    Model answer

    \(\begin{pmatrix} 3 \\ -5 \end{pmatrix} = \begin{pmatrix} 3 \\ -5 \end{pmatrix}\).

    Mark scheme

    • Subtracts the top numbers or the bottom numbers — M1
    • \(\begin{pmatrix} 3 \\ -5 \end{pmatrix}\) — A1
  2. 2 Calculate [4 marks]

    The vectors \(\mathbf{m}\) and \(\mathbf{n}\) are drawn on the grid. (a) Write \(\mathbf{m}\) and \(\mathbf{n}\) as column vectors. [2 marks] (b) Calculate \(\mathbf{m} + 2\mathbf{n}\). [2 marks]

    Two vectors m and n drawn as arrows on a grid.
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    Model answer

    (a) \(\mathbf{m}\) goes 3 right and 1 up, so \(\mathbf{m} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\). \(\mathbf{n}\) goes 2 left and 3 down, so \(\mathbf{n} = \begin{pmatrix} -2 \\ -3 \end{pmatrix}\). (b) \(2\mathbf{n} = \begin{pmatrix} -4 \\ -6 \end{pmatrix}\), so \(\mathbf{m} + 2\mathbf{n} = \begin{pmatrix} -1 \\ -5 \end{pmatrix}\).

    Mark scheme

    • (a) \(\mathbf{m} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\) — B1
    • (a) \(\mathbf{n} = \begin{pmatrix} -2 \\ -3 \end{pmatrix}\) — B1
    • (b) \(2\mathbf{n} = \begin{pmatrix} -4 \\ -6 \end{pmatrix}\) or a correct method — M1
    • (b) \(\begin{pmatrix} -1 \\ -5 \end{pmatrix}\) — A1 (follow through from (a))
  3. 3 Calculate [3 marks]

    \(A\) is the point \((5, -2)\) and \(B\) is the point \((-1, 4)\). (a) Calculate \(\overrightarrow{AB}\) as a column vector. [2 marks] (b) Write down \(\overrightarrow{BA}\) as a column vector. [1 mark]

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    Model answer

    (a) \((-1 - 5, 4 - (-2)) = (-6, 6)\), so \(\overrightarrow{AB} = \begin{pmatrix} -6 \\ 6 \end{pmatrix}\). (b) \(\overrightarrow{BA} = \begin{pmatrix} 6 \\ -6 \end{pmatrix}\).

    Mark scheme

    • (a) \(-1 - 5\) or \(4 - (-2)\) — M1
    • (a) \(\begin{pmatrix} -6 \\ 6 \end{pmatrix}\) — A1
    • (b) \(\begin{pmatrix} 6 \\ -6 \end{pmatrix}\) — B1 (follow through from (a))
  4. 4 Calculate [3 marks]

    \(\mathbf{u} = \begin{pmatrix} 1 \\ -2 \end{pmatrix}\) and \(\mathbf{v} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}\). (a) Calculate \(3\mathbf{u} - \mathbf{v}\). [2 marks] (b) Calculate \(2\mathbf{u} + \mathbf{v}\). [1 mark]

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    Model answer

    (a) \(3\mathbf{u} = \begin{pmatrix} 3 \\ -6 \end{pmatrix}\), so \(3\mathbf{u} - \mathbf{v} = \begin{pmatrix} 0 \\ -11 \end{pmatrix}\). (b) \(2\mathbf{u} = \begin{pmatrix} 2 \\ -4 \end{pmatrix}\), so \(2\mathbf{u} + \mathbf{v} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}\).

    Mark scheme

    • (a) \(3\mathbf{u} = \begin{pmatrix} 3 \\ -6 \end{pmatrix}\) or a correct method — M1
    • (a) \(\begin{pmatrix} 0 \\ -11 \end{pmatrix}\) — A1
    • (b) \(\begin{pmatrix} 5 \\ 1 \end{pmatrix}\) — B1
  5. 5 Show that [2 marks]

    Show that the vectors \(\begin{pmatrix} -4 \\ 6 \end{pmatrix}\) and \(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\) are parallel. [2 marks]

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    Model answer

    \(\begin{pmatrix} -4 \\ 6 \end{pmatrix} = -2 \times \begin{pmatrix} 2 \\ -3 \end{pmatrix}\). One vector is a multiple of the other, so they are parallel.

    Mark scheme

    • \(-2 \times \begin{pmatrix} 2 \\ -3 \end{pmatrix}\) — M1
    • A multiple, so parallel — A1
  6. 6 Calculate [3 marks]

    (a) Calculate the length of the vector \(\begin{pmatrix} 5 \\ 12 \end{pmatrix}\). [2 marks] (b) Calculate the length of the vector \(\begin{pmatrix} 8 \\ 6 \end{pmatrix}\). [1 mark]

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    Model answer

    (a) \(\sqrt{5^2 + 12^2} = \sqrt{169} = 13\). (b) \(\sqrt{8^2 + 6^2} = \sqrt{100} = 10\).

    Mark scheme

    • (a) \(\sqrt{5^2 + 12^2}\) or \(\sqrt{169}\) — M1
    • (a) 13 — A1
    • (b) 10 — B1

Quick check

  1. 1

    What does the column vector \(\begin{pmatrix} -2 \\ 5 \end{pmatrix}\) mean?

    1. A2 right and 5 up
    2. B2 left and 5 up
    3. C5 left and 2 up
    4. D2 left and 5 down
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    B: 2 left and 5 up

    The top number is the horizontal move, and the bottom number is the vertical move.

  2. 2

    \(A\) is \((2, 5)\) and \(B\) is \((6, 2)\). What is \(\overrightarrow{AB}\)?

    1. A\(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\)
    2. B\(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 4 \\ 3 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 8 \\ 7 \end{pmatrix}\)
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    A: \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\)

    Subtract the start from the end: \((6 - 2, 2 - 5) = (4, -3)\).

  3. 3

    What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

    1. A\(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 3 \\ 8 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 10 \\ 10 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\)
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    D: \(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\)

    Add the top numbers and add the bottom numbers.

  4. 4

    What is \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix}\)?

    1. A\(\begin{pmatrix} 4 \\ 6 \end{pmatrix}\)
    2. B\(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 2 \\ 2 \end{pmatrix}\)
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    C: \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)

    \((3 - 1, 2 - 4) = (2, -2)\).

  5. 5

    What is \(3\begin{pmatrix} 2 \\ -1 \end{pmatrix}\)?

    1. A\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 6 \\ -1 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\)
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    B: \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\)

    Multiply both numbers by 3.

  6. 6

    \(\mathbf{p} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}\). What is \(2\mathbf{p} - \mathbf{q}\)?

    1. A\(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 3 \\ 10 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 5 \\ 10 \end{pmatrix}\)
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    A: \(\begin{pmatrix} 5 \\ 2 \end{pmatrix}\)

    \(2\mathbf{p} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\), then \((4 - (-1), 6 - 4) = (5, 2)\).

  7. 7

    Which vector is parallel to \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\)?

    1. A\(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 2 \\ -3 \end{pmatrix}\)
    3. C\(\begin{pmatrix} 4 \\ 5 \end{pmatrix}\)
    4. D\(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\)
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    D: \(\begin{pmatrix} 6 \\ 9 \end{pmatrix}\)

    \(\begin{pmatrix} 6 \\ 9 \end{pmatrix} = 3\begin{pmatrix} 2 \\ 3 \end{pmatrix}\), so it is parallel.

  8. 8

    \(\overrightarrow{AB} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}\). What is \(\overrightarrow{BA}\)?

    1. A\(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\)
    2. B\(\begin{pmatrix} 3 \\ -4 \end{pmatrix}\)
    3. C\(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\)
    4. D\(\begin{pmatrix} -3 \\ 4 \end{pmatrix}\)
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    C: \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\)

    \(\overrightarrow{BA} = -\overrightarrow{AB}\), so both signs change.

  9. 9

    What is the length of the vector \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\)?

    1. A\(7\)
    2. B\(5\)
    3. C\(25\)
    4. D\(12\)
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    B: \(5\)

    The length is \(\sqrt{3^2 + 4^2} = \sqrt{25} = 5\).