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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 Work out [2 marks]

    Work out \(\begin{pmatrix} 3 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ 4 \end{pmatrix}\). [2 marks]

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    Model answer

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

    Mark scheme

    • Adds the top numbers or the bottom numbers — M1
    • \(\begin{pmatrix} 5 \\ 5 \end{pmatrix}\) — A1
  2. 2 Work out [4 marks]

    The vectors \(\mathbf{r}\) and \(\mathbf{s}\) are drawn on the grid. (a) Write \(\mathbf{r}\) and \(\mathbf{s}\) as column vectors. [2 marks] (b) Work out \(3\mathbf{r} - \mathbf{s}\). [2 marks]

    Two vectors r and s drawn as arrows on a grid.
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    Model answer

    (a) \(\mathbf{r}\) goes 2 right and 4 up, so \(\mathbf{r} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\). \(\mathbf{s}\) goes 3 right and 1 down, so \(\mathbf{s} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\). (b) \(3\mathbf{r} = \begin{pmatrix} 6 \\ 12 \end{pmatrix}\), so \(3\mathbf{r} - \mathbf{s} = \begin{pmatrix} 3 \\ 13 \end{pmatrix}\).

    Mark scheme

    • (a) \(\mathbf{r} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\) — B1
    • (a) \(\mathbf{s} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}\) — B1
    • (b) \(3\mathbf{r} = \begin{pmatrix} 6 \\ 12 \end{pmatrix}\) or a correct method — M1
    • (b) \(\begin{pmatrix} 3 \\ 13 \end{pmatrix}\) — A1 (follow through from (a))
  3. 3 Work out [3 marks]

    \(P\) is the point \((2, -1)\) and \(Q\) is the point \((-3, 4)\). (a) Write \(\overrightarrow{PQ}\) as a column vector. [2 marks] (b) Write \(\overrightarrow{QP}\) as a column vector. [1 mark]

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    Model answer

    (a) \((-3 - 2, 4 - (-1)) = (-5, 5)\), so \(\overrightarrow{PQ} = \begin{pmatrix} -5 \\ 5 \end{pmatrix}\). (b) \(\overrightarrow{QP} = \begin{pmatrix} 5 \\ -5 \end{pmatrix}\).

    Mark scheme

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

    \(\mathbf{a} = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} 4 \\ 1 \end{pmatrix}\). (a) Work out \(2\mathbf{a} - \mathbf{b}\). [2 marks] (b) Work out \(\mathbf{a} + \mathbf{b}\). [1 mark]

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    Model answer

    (a) \(2\mathbf{a} = \begin{pmatrix} -4 \\ 10 \end{pmatrix}\), so \(2\mathbf{a} - \mathbf{b} = \begin{pmatrix} -8 \\ 9 \end{pmatrix}\). (b) \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\).

    Mark scheme

    • (a) \(2\mathbf{a} = \begin{pmatrix} -4 \\ 10 \end{pmatrix}\) or a correct method — M1
    • (a) \(\begin{pmatrix} -8 \\ 9 \end{pmatrix}\) — A1
    • (b) \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\) — B1
  5. 5 Write down [2 marks]

    Write down a vector that is parallel to \(\begin{pmatrix} 1 \\ -3 \end{pmatrix}\) and four times as long. [2 marks]

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    Model answer

    Multiply by 4: \(4 \times \begin{pmatrix} 1 \\ -3 \end{pmatrix} = \begin{pmatrix} 4 \\ -12 \end{pmatrix}\).

    Mark scheme

    • Multiplies both numbers by 4 — M1
    • \(\begin{pmatrix} 4 \\ -12 \end{pmatrix}\) — A1
  6. 6 Show that [3 marks]

    \(\overrightarrow{AB} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and \(\overrightarrow{BC} = \begin{pmatrix} 4 \\ 6 \end{pmatrix}\). Show that \(A\), \(B\) and \(C\) lie on a straight line. [3 marks]

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    Model answer

    \(\overrightarrow{BC} = \begin{pmatrix} 4 \\ 6 \end{pmatrix} = 2 \times \begin{pmatrix} 2 \\ 3 \end{pmatrix} = 2\overrightarrow{AB}\). The vectors are parallel and share the point \(B\), so \(A\), \(B\) and \(C\) are on a straight line.

    Mark scheme

    • \(\overrightarrow{BC} = 2\overrightarrow{AB}\) — M1
    • Parallel — A1
    • Common point \(B\), so collinear — Q1

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