[discia] GCSE Maths › Geometry & Measures › Vectors

Vectors

discia.org 10 questions · 26 marks
Name Date
  1. Either paper 1 mark

    Here is a vector, drawn as an arrow from \(A\) to \(B\) on a grid of unit squares.

    AB

    Circle the column vector that represents the arrow from \(A\) to \(B\).

    \(\begin{pmatrix} 3 \\ 2 \end{pmatrix} \qquad \begin{pmatrix} 3 \\ -2 \end{pmatrix} \qquad \begin{pmatrix} -3 \\ 2 \end{pmatrix} \qquad \begin{pmatrix} -3 \\ -2 \end{pmatrix}\)

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    \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\)
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  2. Either paper 2 marks

    The diagram shows the vectors \(\mathbf{a}\) and \(\mathbf{b}\) drawn on a grid of unit squares.

    ab
    (a)

    Write \(\mathbf{a}\) as a column vector.

    (b)

    Write \(\mathbf{b}\) as a column vector.

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    (a) \(\begin{pmatrix} 3 \\ 1 \end{pmatrix}\)
    (b) \(\begin{pmatrix} -4 \\ -2 \end{pmatrix}\)
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  3. Either paper 3 marks

    The points \(A\) and \(B\) are marked on a coordinate grid.

    -5-4-3-2-112345-5-4-3-2-1123450xyAB
    (a)

    Write \(\overrightarrow{AB}\) as a column vector.

    (b)

    Write \(\overrightarrow{BA}\) as a column vector.

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    (a) \(\begin{pmatrix} 4 \\ -6 \end{pmatrix}\)
    (b) \(\begin{pmatrix} -4 \\ 6 \end{pmatrix}\)
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  4. Either paper 1 mark

    The vector \(\begin{pmatrix} -4 \\ 5 \end{pmatrix}\) translates \(C\) to \(D\).

    Write down the vector that translates \(D\) to \(C\).

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    \(\begin{pmatrix} 4 \\ -5 \end{pmatrix}\)
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  5. Either paper 3 marks

    The diagram shows the points \(P\), \(Q\) and \(R\) on a grid of unit squares.

    abPQR

    \(\overrightarrow{PQ} = \mathbf{a}\) and \(\overrightarrow{QR} = \mathbf{b}\).

    (a)

    Write \(\mathbf{a}\) as a column vector.

    (b)

    Write \(\overrightarrow{PR}\) as a column vector.

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    (a) \(\begin{pmatrix} -5 \\ -3 \end{pmatrix}\)
    (b) \(\begin{pmatrix} -8 \\ 0 \end{pmatrix}\)
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  6. Either paper 2 marks

    The diagram shows \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\) on a grid of unit squares.

    abOAB

    Work out \(\mathbf{a} - \mathbf{b}\). Give your answer as a column vector.

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    \(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\)
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  7. Either paper 3 marks

    The vector \(\mathbf{a}\) and the points \(R\) and \(S\) are drawn on a grid of unit squares.

    aRS
    (a)

    Write \(\mathbf{a}\) as a column vector.

    (b)

    Find \(\overrightarrow{RS}\) in terms of \(\mathbf{a}\).

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    (a) \(\begin{pmatrix} 3 \\ -3 \end{pmatrix}\)
    (b) \(3\mathbf{a}\)
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  8. Either paper 4 marks

    The diagram shows the vectors \(\mathbf{a}\) and \(\mathbf{b}\) drawn on a grid of unit squares.

    ab
    (a)

    Work out \(2\mathbf{a}\). Give your answer as a column vector.

    (b)

    Work out \(2\mathbf{a} - \mathbf{b}\). Give your answer as a column vector.

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    (a) \(\begin{pmatrix} 6 \\ 2 \end{pmatrix}\)
    (b) \(\begin{pmatrix} 5 \\ 0 \end{pmatrix}\)
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  9. Either paper 3 marks

    Here is the vector \(\mathbf{a}\) drawn on a grid of unit squares.

    a

    Work out \(|\mathbf{a}|\).

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    \(13\)
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  10. Either paper 4 marks

    \(OPQ\) is a triangle.

    3a3bOPQ

    \(\overrightarrow{OP} = 3\mathbf{a}\) and \(\overrightarrow{OQ} = 3\mathbf{b}\).

    (a)

    Express \(\overrightarrow{PQ}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

    (b)

    Express \(\overrightarrow{QP}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

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    (a) \(3\mathbf{b} - 3\mathbf{a}\)
    (b) \(3\mathbf{a} - 3\mathbf{b}\)
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discia.org Vectors · sheet 1
Answers Vectors · sheet 1
  1. \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\)
  2. (a) \(\begin{pmatrix} 3 \\ 1 \end{pmatrix}\)
    (b) \(\begin{pmatrix} -4 \\ -2 \end{pmatrix}\)
  3. (a) \(\begin{pmatrix} 4 \\ -6 \end{pmatrix}\)
    (b) \(\begin{pmatrix} -4 \\ 6 \end{pmatrix}\)
  4. \(\begin{pmatrix} 4 \\ -5 \end{pmatrix}\)
  5. (a) \(\begin{pmatrix} -5 \\ -3 \end{pmatrix}\)
    (b) \(\begin{pmatrix} -8 \\ 0 \end{pmatrix}\)
  6. \(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\)
  7. (a) \(\begin{pmatrix} 3 \\ -3 \end{pmatrix}\)
    (b) \(3\mathbf{a}\)
  8. (a) \(\begin{pmatrix} 6 \\ 2 \end{pmatrix}\)
    (b) \(\begin{pmatrix} 5 \\ 0 \end{pmatrix}\)
  9. \(13\)
  10. (a) \(3\mathbf{b} - 3\mathbf{a}\)
    (b) \(3\mathbf{a} - 3\mathbf{b}\)
discia.org Vectors · sheet 1