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.

Start practising Read the lesson

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 points $A$ , $B$ and $C$ have coordinates $(1, 2)$ , $(4, 8)$ and $(6, 12)$ .
(a) Find $\overrightarrow{AB}$ and $\overrightarrow{BC}$ . (2 marks)
(b) Show that $A$ , $B$ and $C$ are collinear. (3 marks)
(c) Find the ratio $AB : BC$ . (2 marks)
Level 3 · 7 marksTry it
2
In triangle $OAB$ , $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ . The point $M$ is the midpoint of $AB$ , and $G$ lies on $OM$ with $OG : GM = 2 : 1$ .
(a) Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ . (3 marks)
(b) Show that $\overrightarrow{OG} = \tfrac13(\mathbf{a} + \mathbf{b})$ . (3 marks)
Level 3 · 6 marksTry it
3
The vectors $\mathbf{a} = 3\mathbf{i} + 2\mathbf{j}$ and $\mathbf{b} = 6\mathbf{i} + \lambda\mathbf{j}$ are parallel. Find the value of $\lambda$ . (3 marks)
Level 3 · 3 marksTry it
4
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
5
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

15 more questions on solving geometric problems with vectors

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free