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
Points $A$ and $B$ have position vectors $2\mathbf{i} - \mathbf{j} + \mathbf{k}$ and $4\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ . The point $C$ with position vector $\mathbf{i} + \lambda\mathbf{j} + \mu\mathbf{k}$ lies on the line through $A$ and $B$ . Find $\lambda$ and $\mu$ .
Level 5 · 4 marksTry it
2
The points $A$ , $B$ and $C$ have coordinates $(1, 0, 2)$ , $(5, 8, -2)$ and $(7, 6, k)$ respectively, where $k$ is a constant.
(a) Given that angle $ACB$ is a right angle, find the possible values of $k$ . (3 marks)
(b) Given also that $k > 0$ , find the exact shortest distance from $C$ to the line $AB$ . (3 marks)
(c) The point $D$ is the point on $AB$ that is closest to $C$ . Find the ratio $AD : DB$ in its simplest form. (2 marks)
Level 5 · 8 marksTry it
3
Relative to a fixed origin $O$ , the points $A$ , $B$ and $C$ have position vectors

$\mathbf{a} = -\mathbf{i} + \mathbf{j} + 4\mathbf{k}$ , $\qquad \mathbf{b} = 5\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$ , $\qquad \mathbf{c} = 5\mathbf{i} + 7\mathbf{j} + 4\mathbf{k}$ .
(a) Show that angle $BAC = \dfrac{\pi}{4}$ . (4 marks)

The point $P$ lies on the line segment $AB$ and is such that angle $APC = \dfrac{\pi}{2}$ .
(b) Find the position vector of $P$ . (3 marks)
(c) Find the area of triangle $ABC$ . (2 marks)
Level 5 · 9 marksTry it
4
Figure 2 shows the triangle $OAB$ , where $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$ .

The point $M$ is the midpoint of $OB$ . The point $N$ lies on $AB$ with $AN : NB = 1 : 3$ . The lines $ON$ and $AM$ intersect at the point $X$ .Figure 2 (not accurately drawn)

Given that $\overrightarrow{OX} = \lambda\,\overrightarrow{ON}$ and $\overrightarrow{AX} = \mu\,\overrightarrow{AM}$ , where $\lambda$ and $\mu$ are scalars, find the value of $\lambda$ and the value of $\mu$ . (7 marks)
Level 5 · 7 marksTry it
5
Relative to a fixed origin $O$ , the points $A$ , $B$ and $C$ have position vectors

$\mathbf{a} = \mathbf{i} + 2\mathbf{k}, \qquad \mathbf{b} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k} \qquad \text{and} \qquad \mathbf{c} = 7\mathbf{i} + 6\mathbf{j} - \mathbf{k}.$
(a) Show that angle $ABC$ is a right angle. (3 marks)
(b) The point $D$ lies on $AC$ and is such that $BD$ is perpendicular to $AC$ . Determine the position vector of $D$ . (5 marks)
(c) Find the exact length of $BD$ . (1 mark)
Level 5 · 9 marksTry it

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