A line \(L_1\) makes angles \(\displaystyle\frac{\pi}{3}, \frac{\pi}{4}, \frac{\pi}{3}\) and a line \(L_2\) makes angles \(\displaystyle\frac{2\pi}{3}, \frac{\pi}{4}, \frac{\pi}{3}\) with the positive directions of the x-axis, y-axis and z-axis respectively. The acute angle between \(L_1\) and \(L_2\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{\pi}{3}\)
The direction cosines of a line are the cosines of the angles it makes with the axes:
\( \displaystyle L_1:\ \frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2} \)
\( \displaystyle L_2:\ -\frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2} \)
For direction cosines, \(\cos\theta = |l_1l_2+m_1m_2+n_1n_2|\):
\( \displaystyle \cos\theta = \left|-\frac{1}{4}+\frac{1}{2}+\frac{1}{4}\right| = \frac{1}{2} \)
\( \displaystyle \theta = \frac{\pi}{3} \)