A) \[\frac{\pi }{2}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{4}\]
D) \[\frac{5\pi }{12}\]
Correct Answer: B
Solution :
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 \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\] \[\Rightarrow \] \[\frac{1}{4}+\frac{1}{2}+{{n}^{2}}=1\] \[\Rightarrow \] \[{{n}^{2}}=\frac{1}{4}\Rightarrow n=\frac{1}{2}\] \[\therefore \] \[\cos \gamma =\frac{1}{2}=\cos \frac{\pi }{3}\Rightarrow \gamma =\frac{\pi }{3}\]
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