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
$ 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} $