A-level · Vectors

Position vectorsExam-style questions

Pinning points to the origin with position vectors, the "destination minus start" rule AB = b − a, the distance between two points as |b − a|, the midpoint as ½(a + b), and finding a point that divides a segment in a ratio.

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15 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
$ABCD$ is a quadrilateral. The points $A$ and $B$ have position vectors $\mathbf{i} - 3\mathbf{j}$ and $4\mathbf{i} + \mathbf{j}$ respectively.
(a) Find the length $AB$ .

(1 mark)

The vertex $C$ lies on the positive $y$ -axis, and $BC = AB$ .
(b) Determine the position vector of $C$ .

(3 marks)

The vertex $D$ is such that $\overrightarrow{AD} = 2\overrightarrow{BC}$ .
(c) Determine the position vector of $D$ .

(2 marks)
(d) State, with a reason, the type of quadrilateral that $ABCD$ is.

(2 marks)
Level 4 · 8 marksTry it
2
The points $A$ and $B$ have position vectors $\begin{pmatrix} 4 \\ -1 \\ 8 \end{pmatrix}$ and $\begin{pmatrix} 6 \\ -1 \\ 7 \end{pmatrix}$ respectively. The point $P$ lies on the line through the origin $O$ and $A$ , with $\overrightarrow{OP} = t\overrightarrow{OA}$ , where $t$ is a positive constant.
(a) Write down an expression, in terms of $t$ , for the distance $OP$ .

(1 mark)

It is given that $B$ is equidistant from $O$ and $P$ .
(b) Determine the value of $t$ .

(4 marks)
Level 4 · 5 marksTry it
3
The points $A$ and $B$ in the $x$ - $y$ plane have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a} \ne \mathbf{b}$ .
(a) (i) Write down, in terms of $\mathbf{a}$ and $\mathbf{b}$ , the position vector of the point $C$ for which $A$ is the midpoint of $BC$ .

(ii) Write down an expression, in terms of $\mathbf{a}$ and $\mathbf{b}$ , for the length $BC$ .

(2 marks)

The point $P$ has position vector $\mathbf{p}$ , where $\left|\mathbf{p} - (2\mathbf{a} - \mathbf{b})\right| = |\mathbf{a} - \mathbf{b}|$ .
(b) (i) Describe the locus of $P$ .

(ii) Explain why $A$ lies on this locus.

(2 marks)
(c) In the case where $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$ , find a cartesian equation of the locus of $P$ .

(4 marks)
Level 4 · 8 marksTry it
4
Points $A$ and $B$ have position vectors $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ and $5\mathbf{i} + 4\mathbf{j} + 14\mathbf{k}$ . Find the position vector of the midpoint $M$ of $AB$ , and the distance $AM$ .
Level 4 · 4 marksTry it
5
The points $A$ , $B$ and $C$ have position vectors, relative to the origin $O$ ,

$\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \qquad \overrightarrow{OB} = 4\mathbf{i} + 6\mathbf{j} + 3\mathbf{k}, \qquad \overrightarrow{OC} = \mathbf{i} + 5\mathbf{j} + 7\mathbf{k}$
(a) Find $\overrightarrow{AB}$ and $\overrightarrow{AC}$ , and show that the triangle $ABC$ is isosceles. (4 marks)
(b) Find the exact length of $BC$ . (2 marks)
(c) Use the cosine rule to find angle $BAC$ , correct to $1$ decimal place. (3 marks)
Level 4 · 9 marksTry it

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