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
$OAB$ is a triangle. $\overrightarrow{OA} = 2\mathbf{a}$ and $\overrightarrow{OB} = 2\mathbf{b}$ .

$M$ is the midpoint of $OA$ and $N$ is the midpoint of $OB$ .Diagram not accurately drawn.

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

(b) Prove that $MN$ is parallel to $AB$ .

(c) Hence write down the ratio $MN : AB$ .
Level 4 · 6 marksTry it
2
$PQRS$ is a quadrilateral.

$\overrightarrow{PQ} = 5\mathbf{a} + 2\mathbf{b}$ , $\quad\overrightarrow{QR} = 3\mathbf{a} - 4\mathbf{b}$ , $\quad\overrightarrow{RS} = -5\mathbf{a} - 2\mathbf{b}$ .

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

(b) Prove that $PQRS$ is a parallelogram.
Level 4 · 5 marksTry it
3
$O$ , $A$ and $B$ are three points, with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .

The point $C$ is such that $\overrightarrow{OC} = \tfrac{1}{5}\mathbf{a} + \tfrac{4}{5}\mathbf{b}$ .

(a) Show that $C$ lies on the line $AB$ .

(b) Write down the ratio $AC : CB$ .
Level 4 · 4 marksTry it
4
$ABCD$ is a quadrilateral.

$\overrightarrow{AB} = 4\mathbf{a} + 2\mathbf{b}$ , $\quad\overrightarrow{BC} = \mathbf{a} - 3\mathbf{b}$ , $\quad\overrightarrow{CD} = -2\mathbf{a} - 8\mathbf{b}$ .

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

(b) Show that $AD$ is parallel to $BC$ .

(c) Aisha says that $ABCD$ is a parallelogram. Aisha is wrong. Explain why.
Level 4 · 5 marksTry it