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 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