question_answer

If a unit vector \[\hat{a}\] makes angles \[\frac{\pi }{3}\] with \[\hat{i},\,\,\frac{\pi }{4}\]  with \[\hat{j}\] and an acute angle \[\gamma \] with \[\hat{k}\] find the value of \[\gamma .\]

Answer:

We know that, \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1\] Here,                 \[\alpha =\frac{\pi }{3}\]and \[\beta =\frac{\pi }{4}\] So,       \[{{\cos }^{2}}\frac{\pi }{3}+{{\cos }^{2}}\frac{\pi }{4}+{{\cos }^{2}}\gamma =1\] \[\Rightarrow \]   \[{{\left( \frac{1}{2} \right)}^{2}}+{{\left( \frac{1}{\sqrt{2}} \right)}^{2}}+{{\cos }^{2}}\gamma =1\]      \[\Rightarrow \]   \[\frac{3}{4}+{{\cos }^{2}}\gamma =1\] \[\Rightarrow \]   \[{{\cos }^{2}}\gamma =1-\frac{3}{4}=\frac{1}{4}\Rightarrow \cos \gamma =\frac{1}{2}\]         \[\therefore \]      \[\gamma =\frac{\pi }{3}\]