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Question
Find the value of θ ∈(0, π/2) for which vectors \[\vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \text{ and } \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k}\] are perpendicular.
Solution
\[\text{ We have }\]
\[ \vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \]
\[\text{ and }\]
\[ \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k} \]
\[\text{ It is given that the vectors are perpendicular }.\]
\[ \Rightarrow \vec{a} . \vec{b} = 0\]
\[ \Rightarrow \sin \theta - \sqrt{3} \cos \theta = 0\]
\[ \Rightarrow \sin \theta = \sqrt{3} \cos \theta\]
\[ \Rightarrow \tan \theta = \sqrt{3} \]
\[ \Rightarrow \theta = \frac{\pi}{3}\]
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