Q. If a line makes an angle of $\frac{\pi}{4}$ with the positive direction of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $\mathcal{z}$-axis is
A) $\frac{\pi}{3}$
B) $\frac{\pi}{4}$
C) $\frac{\pi}{2}$
D) $\frac{\pi}{6}$
Explanation
(c)
Here $l = \cos\frac{\pi}{4},\ m = \cos\frac{\pi}{4}$
Let the line make an angle ‘$\gamma$’ with $z$-axis
$\therefore l^{2} + m^{2} + n^{2} = 1$
$\Rightarrow \cos^{2}\frac{\pi}{4} + \cos^{2}\frac{\pi}{4} + \cos^{2}\gamma = 1$
$\Rightarrow \frac{1}{2} + \frac{1}{2} + \cos^{2}\gamma = 1$
$\Rightarrow 2\cos^{2}\gamma = 0\ \Rightarrow \cos\gamma = 0\ \Rightarrow \ \gamma = \frac{\pi}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.