Question

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 $z$-axis is-

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (C)

$\frac{\pi }{2}$

let $\alpha$, $\beta$ and $\gamma$ be the angle made by the line with co-ordinate axes.
Therefore, ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\frac{\pi }{4}+{\cos }^2\frac{\pi }{4}+{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\gamma =0$

$\Rightarrow \gamma =\frac{\pi }{2}$
Ans: C