Question

If a line makes an angle of $\pi /3$ with each of $x$ and and $y$-axis, then the acute angle made by $z$-axis is

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

Answer 1

Answer: (A)

Given $\alpha =\beta =\pi /3$
Let acute angle made by $z$ -axis be $y$
Then, $co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma$
$\Rightarrow (\frac{1}{2}{)}^2+(\frac{1}{2}{)}^2+co{s}^2\gamma =1$
$\Rightarrow \frac{1}{4}+\frac{1}{4}+co{s}^2\gamma =1$
$\Rightarrow co{s}^2\gamma =\frac{1}{2}$
$\Rightarrow cos\gamma =\pm \frac{1}{\sqrt{2}}$
$\Rightarrow \gamma =\frac{\pi }{4}$
$[\therefore y$ is acute]