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)

$\frac{\pi }{3}$

Answer 2

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