WAEC 2022 · Paper 2 · Q15

Vectors

The vectors $6\mathbf i + 8\mathbf j$ and $8\mathbf i - 6\mathbf j$ are parallel to $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ respectively. If the magnitudes of $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ are 80 units and 120 units respectively, express:

(a)
$\overrightarrow{OP}$ and $\overrightarrow{OQ}$ in terms of $\mathbf i$ and $\mathbf j$ ;
Show the answer
$\overrightarrow{OP} = 48\mathbf i + 64\mathbf j$ , $\overrightarrow{OQ} = 96\mathbf i - 72\mathbf j$
(b)
$|\overrightarrow{PQ}|$ in the form $c\sqrt k$ , where $c$ and $k$ are constants.

Worked solution (try it first)

(a)

$|6\mathbf i + 8\mathbf j| = 10$ , so $\overrightarrow{OP} = \frac{80}{10}(6\mathbf i + 8\mathbf j)$
$= 48\mathbf i + 64\mathbf j$ .
$|8\mathbf i - 6\mathbf j| = 10$ , so $\overrightarrow{OQ} = \frac{120}{10}(8\mathbf i - 6\mathbf j)$
$= 96\mathbf i - 72\mathbf j$ .

(b)

$\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP}$
$= 48\mathbf i - 136\mathbf j$ .
$|\overrightarrow{PQ}| = \sqrt{2304 + 18\,496}$
$= \sqrt{20\,800}$ .
$20\,800 = 1600 \times 13$ , so $|\overrightarrow{PQ}| = 40\sqrt{13}$ .

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