A-level · Vectors

Magnitude and direction of a vectorExam-style questions

The magnitude of a vector as Pythagoras on its components (left as an exact surd), unit vectors as a vector scaled to length one, direction as an angle or bearing, and finding the angle in a vector triangle with the cosine rule.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The point $A$ lies in the fourth quadrant, at a distance of $12$ from the origin $O$ . The line $OA$ makes an angle of $60°$ with the positive $x$ -axis.
(a) (i) Show the position of $A$ on a sketch of the $x$ - and $y$ -axes.

(ii) Find the position vector of $A$ , giving your answer in terms of $\mathbf{i}$ and $\mathbf{j}$ .

(4 marks)

The point $B$ lies on the negative $x$ -axis, and $OB = 8$ .
(b) Find the exact area of triangle $OAB$ .

(2 marks)
(c) Find the position vector of the point $D$ for which $OADB$ is a parallelogram, giving your answer in terms of $\mathbf{i}$ and $\mathbf{j}$ .

(2 marks)
Level 4 · 8 marksTry it
2
Given $\mathbf{a} = 4\mathbf{i} + \mathbf{j} - 2\mathbf{k}$ and $\mathbf{b} = \mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ , find $|\mathbf{a} - \mathbf{b}|$ in surd form.
Level 4 · 3 marksTry it
3
The points $A$ and $B$ have position vectors $\mathbf{a} = 2\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ and $\mathbf{b} = 4\mathbf{i} + 3\mathbf{j} - \mathbf{k}$ .
(a) Find $\left|\overrightarrow{AB}\right|$ . (2 marks)
(b) Write down the unit vector in the direction of $\overrightarrow{AB}$ . (1 mark)
(c) The point $C$ has position vector $\mathbf{c} = c\,\mathbf{j} + 5\mathbf{k}$ , where $c$ is a constant. Given that $\left|\overrightarrow{AC}\right| = \left|\overrightarrow{AB}\right|$ , find the possible values of $c$ , giving your answers in exact form. (3 marks)
(d) The point $P$ lies on $AB$ with $AP : PB = 3 : 1$ . Find the position vector of $P$ . (2 marks)
Level 4 · 8 marksTry it