A-level · Vectors

Solving geometric problems with vectorsExam-style questions

Using vectors to prove geometry: parallel vectors as scalar multiples, collinear points as parallel vectors through a shared point, dividing a line in a ratio, expressing figures in terms of base vectors, and comparing areas.

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

DifficultyAll Level 3 Level 4 Level 5

1
The diagram shows triangle $OAB$ , where $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ . The point $C$ lies on $AB$ with $AC : CB = 2 : 1$ , and $D$ is the midpoint of $OC$ . The line through $A$ and $D$ meets $OB$ at the point $E$ .(a) Find $\overrightarrow{AD}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ .

(2 marks)
(b) Explain why $\overrightarrow{AE} = k(2\mathbf{b} - 5\mathbf{a})$ for some constant $k$ .

(2 marks)
(c) Hence determine the ratio $OE : EB$ .

(4 marks)
Level 4 · 8 marksTry it
2
The diagram shows a cuboid $OABCDEFG$ . The vertices $D$ , $E$ , $F$ and $G$ lie vertically above $O$ , $A$ , $B$ and $C$ respectively. $\overrightarrow{OA} = \mathbf{a}$ , $\overrightarrow{OC} = \mathbf{c}$ and $\overrightarrow{OD} = \mathbf{d}$ .The point $P$ lies on the edge $DG$ with $DP : PG = 1 : 3$ , and the point $Q$ lies on the edge $AB$ with $AQ : QB = 3 : 1$ .
(a) Find, in terms of $\mathbf{a}$ , $\mathbf{c}$ and $\mathbf{d}$ , the position vector of the midpoint of $PQ$ .

(4 marks)
(b) Hence show that the line segments $PQ$ and $OF$ bisect each other.

(2 marks)
Level 4 · 6 marksTry it
3
The points $A$ , $B$ , $C$ have position vectors $\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}$ , $3\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$ and $7\mathbf{i} + 5\mathbf{j} + 9\mathbf{k}$ . Show that $A$ , $B$ and $C$ are collinear.
Level 4 · 2 marksTry it
4
$OACB$ is a parallelogram, with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ , so that $\overrightarrow{OC} = \mathbf{a} + \mathbf{b}$ .(a) Prove that the diagonals $OC$ and $AB$ bisect each other. (4 marks)
(b) Given that $\mathbf{a} = 2\mathbf{i} + 3\mathbf{j}$ and $\mathbf{b} = 4\mathbf{i} - \mathbf{j}$ , find the exact lengths of the two diagonals. (4 marks)
Level 4 · 8 marksTry it
5
Relative to a fixed origin $O$ , the points $A$ and $B$ have position vectors $\mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$ and $3\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ respectively.

The point $P$ lies on the straight line through $A$ and $B$ , and is the point on this line that is closest to $O$ .
(a) Find the position vector of $P$ . (4 marks)
(b) Hence find the exact area of triangle $OAB$ . (3 marks)
Level 4 · 7 marksTry it

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