WAEC 2018 · Paper 2 · Q15

(a)
If $\mathbf a = 6\mathbf i - 5\mathbf j$ , $\mathbf b = 2\mathbf i + 7\mathbf j$ and $\mathbf c = 8\mathbf i + 2\mathbf j$ , find the values of the scalars $x$ and $y$ such that $\mathbf c = x\mathbf a + y\mathbf b$ .
Separate values with commas, e.g. 3, −2
(b)
The vectors $\mathbf a$ and $\mathbf b$ are such that $|\mathbf a| = 3\text{ cm}$ , $|\mathbf b| = 10\text{ cm}$ and $|\mathbf a + \mathbf b| = \sqrt{139}$ . Find: (i) the angle between $\mathbf a$ and $\mathbf b$ ; (ii) the scalar (dot) product of $\mathbf a$ and $\mathbf b$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

$|\mathbf a + \mathbf b|^2 = |\mathbf a|^2 + |\mathbf b|^2 + 2|\mathbf a||\mathbf b|\cos\theta$ .
$139 = 9 + 100 + 60\cos\theta$ , so $\cos\theta = \frac{30}{60} = \frac12$ and $\theta = 60^\circ$ .

$\mathbf a \cdot \mathbf b = |\mathbf a||\mathbf b|\cos\theta$
$= 3 \times 10 \times \frac12$
$= 15$ .