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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.
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Solution
\[ \text { Since a unit vector makes an angle of } \frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text { with } \hat {j} \text{ and an acute angle } \theta \text{ with } \hat{k} , l = \cos \frac{\pi}{3} \text { or } \frac{1}{2}, m = \cos \frac{\pi}{4}\text { or } \frac{1}{\sqrt{2}} \text { and } n = \cos \theta . \]
\[\text{ We know } \]
\[ l^2 + m^2 + n^2 = 1\]
\[ \Rightarrow \left( \frac{1}{2} \right)^2 + \left( \frac{1}{\sqrt{2}} \right)^2 + \cos^2 \theta = 1\]
\[ \Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \theta = 1 \]
\[ \Rightarrow \cos^2 \theta = \frac{1}{4}\]
\[ \Rightarrow \cos \theta = \frac{1}{2} \]
\[ \Rightarrow \theta = \frac{\pi}{3}\]
\[\text { Thus, the vector } \vec{a} \text { makes an angle of } \frac{\pi}{3} \text { with } \hat {k} .\]
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