GCSE · Vectors

Column vectorsExam-style questions

A vector is a translation — so far across, so far up — not a place. How to add, subtract and scale columns, why a scalar multiple is parallel, how Pythagoras gives the magnitude, and why the journey from A to B is destination minus start.

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16 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
$\mathbf{a} = \binom{5}{-2}$ and $\mathbf{b} = \binom{-3}{4}$ .

(a) Work out $\mathbf{a} + \mathbf{b}$ as a column vector.

(b) Work out $\mathbf{a} - \mathbf{b}$ as a column vector.

(c) Work out $3\mathbf{b}$ as a column vector.
Level 2 · 3 marksTry it
2
$\mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$

(a) Work out $\mathbf{a} - \mathbf{b}$ (1 mark)

(b) Work out $3\mathbf{a} + 2\mathbf{b}$ (1 mark)

(c) $P$ is the point $(1, 2)$ and $Q$ is the point $(4, 8)$ . Write $\overrightarrow{PQ}$ as a column vector. (1 mark)
Level 2 · 3 marksTry it
3
$\mathbf{a} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -1 \\ 4 \end{pmatrix}$
(a) Work out $2\mathbf{a} + \mathbf{b}$ as a column vector. (1 mark)
(b) Work out the magnitude of $\mathbf{a}$ . Give your answer correct to 3 significant figures. (2 marks)
Level 2 · 3 marksTry it
4
$\mathbf{a} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$

(a) Work out $2\mathbf{a} - \mathbf{b}$ [1]

(b) Write down the exact magnitude of $\mathbf{a}$ . [1]

(c) $P$ is $(-1, 2)$ and $Q$ is $(3, -4)$ . Write $\overrightarrow{PQ}$ as a column vector. [1]

(d) Show that $\begin{pmatrix} 6 \\ 2 \end{pmatrix}$ is parallel to $\mathbf{a}$ . [1]
Level 2 · 4 marksTry it