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
$P$ , $Q$ , $R$ and $S$ are four points. $\overrightarrow{PQ} = 2\mathbf{a} + 6\mathbf{b}$ and $\overrightarrow{RS} = 3\mathbf{a} + 9\mathbf{b}$ .

(a) Show that $PQ$ is parallel to $RS$ .

(b) Freya says: "This working also proves that $P$ , $Q$ , $R$ and $S$ all lie on the same straight line."

Freya is wrong. Explain why.
Level 3 · 3 marksTry it
2
$A$ , $B$ and $C$ are three points.

$\overrightarrow{AB} = 4\mathbf{a} - 2\mathbf{b}$ and $\overrightarrow{BC} = 10\mathbf{a} - 5\mathbf{b}$ .

Prove that $A$ , $B$ and $C$ are collinear.
Level 3 · 3 marksTry it
3
$OAB$ 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
4
$PQRS$ 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
5
$O$ , $A$ 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

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