WAEC 2016 · Paper 2 · Q14

Vectors

Four vectors, $\mathbf r = \alpha\mathbf i + \beta\mathbf j$ where $\alpha$ and $\beta$ are positive constants, $\mathbf s = 2\mathbf i - \mathbf j$ , $\mathbf m = 3\mathbf i + 2\mathbf j$ and $\mathbf n = \mathbf i + \mathbf j$ , are such that the magnitude of $\mathbf r$ is three times that of $\mathbf s$ and $\mathbf r$ is parallel to $(\mathbf m - \mathbf n)$ .

(a)
Find the values of $\alpha$ and $\beta$ .
Separate values with commas, e.g. 3, −2
(b)
Calculate the magnitude and direction (bearing) of $(\mathbf r - \mathbf s)$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

$\mathbf m - \mathbf n = 2\mathbf i + \mathbf j$ , so $\mathbf r = k(2\mathbf i + \mathbf j)$ for some $k > 0$ (the parts are positive).
$|\mathbf s| = \sqrt{4 + 1} = \sqrt5$ , so $|\mathbf r| = 3\sqrt5$ .
$|\mathbf r| = k\sqrt5 = 3\sqrt5$ , so $k = 3$ : $\alpha = 6$ and $\beta = 3$ .

(b)

$\mathbf r - \mathbf s = (6 - 2)\mathbf i + (3 + 1)\mathbf j$
$= 4\mathbf i + 4\mathbf j$ .
Its magnitude is $\sqrt{32} = 4\sqrt2 \approx 5.66$ .
Equal east and north parts: the bearing is $045^\circ$ .

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