GCSE Maths revision · Geometry and measures

Column vectors

Foundation and Higher

The idea

A column vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ means $3$ right and $2$ down. Add vectors by adding the tops and the bottoms; multiply by a number by multiplying both.

Top number: across (right is positive). Bottom number: up or down (up is positive).
Add or subtract the matching components.
Going the opposite way reverses both signs.

Watch out $\overrightarrow{BA} = -\overrightarrow{AB}$ : reversing a vector changes both signs.

See it worked

Worked exampleNon-calculator4 marks

$A$ is the point $(1, 4)$ and $B$ is the point $(5, 1)$ .

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

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

(c) $\overrightarrow{AC} = 2\overrightarrow{AB}$ . Find the coordinates of $C$ .

(a) From $A$ to $B$ : $5 - 1 = 4$ across and $1 - 4 = -3$ up.
(b) Going the other way reverses both signs: $\overrightarrow{BA} = -\overrightarrow{AB}$ .
(c) $2\overrightarrow{AB} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}$
Start at $A(1, 4)$ : $C = (1 + 8, 4 - 6) = (9, -2)$ .

Answer (a) $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$ ; (b) $\begin{pmatrix} -4 \\ 3 \end{pmatrix}$ ; (c) $(9, -2)$

Your turn

Question 1Non-calculator4 marks

$\mathbf{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$ .

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

(b) Work out $3\mathbf{a}$ as a column vector.

Question 2Non-calculator4 marks

A particle is translated by $(4,-3)$ , then by $(-2,5)$ .

(a) Find the resultant translation.

(b) Find the single translation that returns the particle to its starting point.

Question 3Non-calculator5 marks

$p\begin{pmatrix} 2 \\ 1 \end{pmatrix} + q\begin{pmatrix} 1 \\ -3 \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \end{pmatrix}$ , where $p$ and $q$ are numbers.

(a) Find the values of $p$ and $q$ . Give your answer as $(p, q)$ .

(b) The vector $\begin{pmatrix} k \\ 6 \end{pmatrix}$ is parallel to $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ . Find the value of $k$ .

In past papers

72 official questions in 71 papers since 2017. The latest:

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