NECO 2023 · Paper 2 · Q3

Vectors

Given that $\mathbf{x} = 2\mathbf{i} - 5\mathbf{j}$ and $\mathbf{y} = 4\mathbf{i} + 3\mathbf{j}$ , find the:

(i)
angle between them (degrees, 1 d.p.);
(ii)
unit vector in the direction of $5\mathbf{x} + 2\mathbf{y}$ .
Show the answer
$\dfrac{1}{\sqrt{685}}(18\mathbf{i} - 19\mathbf{j})$

Worked solution (try it first)

(i)

$\mathbf x \cdot \mathbf y = (2)(4) + (-5)(3)$
$= 8 - 15$
$= -7$ .
$|\mathbf x| = \sqrt{4 + 25} = \sqrt{29}$ and $|\mathbf y| = \sqrt{16 + 9} = 5$ .
$\cos\theta = \dfrac{-7}{5\sqrt{29}}$
$\approx -0.2600$ , so $\theta \approx 105.1^\circ$ .

(ii)

$5\mathbf x + 2\mathbf y = (10 + 8)\mathbf i + (-25 + 6)\mathbf j$
$= 18\mathbf i - 19\mathbf j$ .
Its length is $\sqrt{324 + 361} = \sqrt{685}$ .
Unit vector: $\dfrac{1}{\sqrt{685}}(18\mathbf i - 19\mathbf j)$ .

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