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
Find $|2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}|$ .
Level 2 · 2 marksTry it
2
The column vector $\begin{pmatrix} a \\ b \end{pmatrix}$ , in which $a$ and $b$ are both positive, has length $10\sqrt{2}$ and is inclined at $30°$ to the positive $x$ -axis.

Determine the exact values of $a$ and $b$ .

(4 marks)
Level 3 · 4 marksTry it
3
Figure 1 shows a sketch of triangle $ABC$ (not accurately drawn). Given that $\vec{AB} = -3\mathbf{i} - 4\mathbf{j} - 5\mathbf{k}$ and $\vec{BC} = \mathbf{i} + \mathbf{j} + 4\mathbf{k}$ , (a) find $\vec{AC}$ , (b) show that $\cos(\angle ABC) = \dfrac{9}{10}$ .
Level 3 · 5 marksTry it
4
The vector $\mathbf{v} = 5\mathbf{i} - 12\mathbf{j}$ .
(a) Find $|\mathbf{v}|$ . (2 marks)
(b) Find the unit vector in the direction of $\mathbf{v}$ . (2 marks)
(c) Find the angle, to 1 d.p., that $\mathbf{v}$ makes with the vector $\mathbf{i}$ . (3 marks)
Level 3 · 7 marksTry it
5
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

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