WAEC 2019 · Paper 2 · Q7

(a)
Find the angle between $\overrightarrow{OP} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$ and $\overrightarrow{OQ} = \begin{pmatrix} 8 \\ -15 \end{pmatrix}$ .

Worked solution (try it first)

$\overrightarrow{OP} \cdot \overrightarrow{OQ} = (-3)(8) + (-4)(-15)$
$= -24 + 60$
$= 36$ .
$|\overrightarrow{OP}| = 5$ and $|\overrightarrow{OQ}| = \sqrt{64 + 225}$
$= 17$ .
$\cos\theta = \dfrac{36}{85}$
$\approx 0.4235$ , so $\theta \approx 64.94^\circ$ .