Question

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

Answer 1

\[ \text { Since a unit vector makes an angle of } \frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text { with } \hat {j}  \text{ and an acute angle }  \theta \text{ with } \hat{k}  , l = \cos \frac{\pi}{3} \text { or } \frac{1}{2}, m = \cos \frac{\pi}{4}\text { or } \frac{1}{\sqrt{2}} \text { and } n = \cos \theta . \]

\[\text{ We know } \]

\[ l^2 + m^2 + n^2 = 1\]

\[ \Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{\sqrt{2}} \right)^2 + \cos^2 \theta = 1\]

\[ \Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \theta = 1 \]

\[ \Rightarrow \cos^2 \theta = \frac{1}{4}\]

\[ \Rightarrow \cos \theta = \frac{1}{2} \]

\[ \Rightarrow \theta = \frac{\pi}{3}\]

\[\text { Thus, the vector }  \vec{a} \text { makes an angle of } \frac{\pi}{3} \text { with }  \hat {k}  .\]