Question

If a line makes angles $\frac{\pi }{3}$ and $\frac{\pi }{4}$ with the $x$ and $y$-axis respectively, then the angle made by the line and $Z$-axis is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{5\pi }{12}$

Answer 1

Answer: (B)

Let $\alpha ,\beta ,\gamma$
be the angles with X-axis, Y-axis, Z-axis respectively, then direction cosines are $\cos \alpha ,\cos \beta <br>$ and $\cos \gamma$
Given, $\alpha =\frac{\pi }{3},\beta =\frac{\pi }{4}$
$\therefore$ $l=\cos \frac{\pi }{3}=\frac{1}{2};m=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}$
and $n=\cos \gamma <br>$ We know that $l^2+m^2+n^2=1$
$\Rightarrow$ $\frac{1}{4}+\frac{1}{2}+n^2=1$
$\Rightarrow$ $n^2=\frac{1}{4}\Rightarrow n=\frac{1}{2}$
$\therefore$ $\cos \gamma =\frac{1}{2}=\cos \frac{\pi }{3}\Rightarrow \gamma =\frac{\pi }{3}$