WAEC 2018 · Paper 2 · Q15

Vectors

(a)
Given that $\mathbf m = 6\mathbf i + 4\mathbf j$ and $\mathbf n = 3\mathbf i - 4\mathbf j$ , find, correct to the nearest degree, the angle between $\mathbf m$ and $\mathbf n$ .
(b)
$A(-1, 1)$ , $B(1, 3)$ , $C(x, y)$ and $D(3, -3)$ are the vertices of a parallelogram $ABCD$ . Calculate the values of $x$ and $y$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

$\mathbf m \cdot \mathbf n = 18 - 16 = 2$ , $|\mathbf m| = \sqrt{52} = 2\sqrt{13}$ and $|\mathbf n| = 5$ .
$\cos\theta = \dfrac{2}{10\sqrt{13}}$
$\approx 0.0555$ , so $\theta \approx 87^\circ$ .

(b)

In $ABCD$ , $\overrightarrow{AB} = \overrightarrow{DC}$ : $\begin{pmatrix} 2 \\ 2 \end{pmatrix} = \begin{pmatrix} x - 3 \\ y + 3 \end{pmatrix}$ .
So $x = 5$ and $y = -1$ .

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