The angle between two lines whose direction ratios are proportional t…

The correct answer is Option (1) → \(\displaystyle\frac{\pi}{3}\)

For direction ratios \(a_1, b_1, c_1\) and \(a_2, b_2, c_2\), the angle between the lines is given by

    \( \displaystyle \cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}} \)

Numerator:

    \(1\left(\sqrt{3}-1\right) + 1\left(-\sqrt{3}-1\right) + (-2)(-4) = -2 + 8 = 6\)

Magnitudes:

    \(\sqrt{1 + 1 + 4} = \sqrt{6}\)

    \(\sqrt{\left(4 - 2\sqrt{3}\right) + \left(4 + 2\sqrt{3}\right) + 16} = \sqrt{24} = 2\sqrt{6}\)

So

    \( \displaystyle \cos\theta = \frac{6}{\sqrt{6}\times 2\sqrt{6}} = \frac{6}{12} = \frac{1}{2} \)

    \( \displaystyle \theta = \frac{\pi}{3} \)