If a line makes angle \(\displaystyle\frac{\pi}{3}\) and \(\displaystyle\frac{\pi}{4}\) with the positive directions of x-axis and y-axis respectively, then the acute angle made by the line with positive direction of z-axis is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{\pi}{3}\)
Let l, m, n be the direction cosines of the line, and \(\gamma\) its angle with the z-axis. Then
\(\displaystyle l = \cos\frac{\pi}{3} = \frac{1}{2},\quad m = \cos\frac{\pi}{4} = \frac{1}{\sqrt{2}},\quad n = \cos\gamma\)
Using \(l^2 + m^2 + n^2 = 1\):
\(\displaystyle \frac{1}{4} + \frac{1}{2} + n^2 = 1\)
\(\displaystyle n^2 = \frac{1}{4},\quad n = \pm\frac{1}{2}\)
\(n = \frac{1}{2}\) gives \(\gamma = \frac{\pi}{3}\), and \(n = -\frac{1}{2}\) gives \(\gamma = \frac{2\pi}{3}\). The acute angle is \(\frac{\pi}{3}\).