GCSE · Vectors

Vector proofsExam-style questions

What a vector equation actually proves — a scalar multiple gives parallel and nothing more, a multiple plus a shared point gives collinear — and how to write the conclusion in the form a mark scheme rewards.

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

DifficultyAll Level 3 Level 4 Level 5

1
$OABC$ is a parallelogram, with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{b}$ .

$M$ is the midpoint of $AB$ .

$N$ is the point on the diagonal $OB$ such that $\overrightarrow{ON} = \tfrac{2}{3}\overrightarrow{OB}$ .Diagram not accurately drawn.

(a) Find $\overrightarrow{CN}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ . Give your answer in its simplest form.

(b) Prove that $C$ , $N$ and $M$ lie on the same straight line.
Level 5 · 7 marksTry it
2
$OAB$ is a triangle, with $\overrightarrow{OA} = 6\mathbf{a}$ and $\overrightarrow{OB} = 6\mathbf{b}$ .

$M$ is the midpoint of $OA$ .

$N$ is the point on $AB$ such that $AN : NB = 2 : 1$ .

$P$ is the point on $OB$ extended such that $\overrightarrow{OP} = p\,\mathbf{b}$ , where $p$ is a number.

(a) Find $\overrightarrow{MN}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ . Give your answer in its simplest form.

(b) $M$ , $N$ and $P$ lie on the same straight line.

Work out the value of $p$ .
Level 5 · 6 marksTry it
3
$OACB$ is a parallelogram.

$\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$

$M$ is the midpoint of $AC$ and $N$ is the midpoint of $BC$ .Diagram NOT accurately drawn

Prove that $MN$ is parallel to $AB$ . (4 marks)
Level 5 · 4 marksTry it
4
$OAB$ is a triangle. $\overrightarrow{OA} = 2\mathbf{a}$ and $\overrightarrow{OB} = 2\mathbf{b}$ .

$P$ is the point on $AB$ such that $AP : PB = 1 : 3$ . $Q$ is the point on $OB$ such that $OQ : QB = 1 : 3$ .

Prove that $PQ$ is parallel to $OA$ . [5]
Level 5 · 5 marksTry it
5
$ABCD$ is a parallelogram. $\overrightarrow{AB} = 4\mathbf{a}$ $\qquad$ $\overrightarrow{AD} = 6\mathbf{b}$

$E$ is the point on $AB$ such that $AE : EB = 1 : 3$

$F$ is the midpoint of the diagonal $AC$ .

The side $BC$ is extended to the point $G$ , where $\overrightarrow{BG} = k\overrightarrow{BC}$

$E$ , $F$ and $G$ lie on a straight line.

Find the value of $k$ .

You must show all your working.

(5 marks)
Level 5 · 5 marksTry it