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)

$\frac{\pi}{4}$

Answer 2

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