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 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
2
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
3
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