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The angle between the lines whose direction ratios are proportional to
$1, 1, 2$ and $\sqrt{3} - 1, - \sqrt{3} - 1, 4$ is

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$60^{\circ}$

$30^{\circ}$

$45^{\circ}$

$90^{\circ}$

answer is A.

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Detailed Solution

Let $θ$ be the angle between the given lines. Then,

$\begin{array}{l} \cos θ = \frac{1 \times (\sqrt{3} - 1) + 1 \times (- \sqrt{3} - 1) + 2 \times 4}{\sqrt{1 + 1 + 4} \sqrt{(\sqrt{3} - 1)^{2} + (- \sqrt{3} - 1)^{2} + 4}} = \frac{1}{\sqrt{2}} \\ \Rightarrow θ = 45^{\circ} \end{array}$

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