If a line makes angle $\frac{\pi}{3}$ and $\frac{\pi}{4}$ 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) → $\frac{\pi}{3}$
Let the direction cosines of the line be $l, m, n$ corresponding to the x, y, and z axes respectively.
Given: $\alpha = \frac{\pi}{3}$, $\beta = \frac{\pi}{4}$
$l = \cos \alpha = \frac{1}{2}$, $m = \cos \beta = \frac{1}{\sqrt{2}}$
Using $l^{2} + m^{2} + n^{2} = 1$:
$\left(\frac{1}{2}\right)^{2} + \left(\frac{1}{\sqrt{2}}\right)^{2} + n^{2} = 1$
$\frac{1}{4} + \frac{1}{2} + n^{2} = 1$
$n^{2} = \frac{1}{4} \Rightarrow n = \frac{1}{2}$
Hence, the angle with the positive z-axis is:
$\gamma = \cos^{-1}\!\left(\frac{1}{2}\right) = \frac{\pi}{3}$
Final Answer: $\displaystyle \frac{\pi}{3}$