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.

Answer 1

\[\text{ We have }\]

\[ \vec{a} = \left( \sin \theta \right) \hat{i} + \left( \cos \theta \right) \hat{j} \]

\[\text{ and }\]

\[ \vec{b} = \hat{i} - \sqrt{3} \hat{j} + 2 \hat{k} \]

\[\text{ It is given that the vectors are perpendicular }.\]

\[ \Rightarrow \vec{a} . \vec{b} = 0\]

\[ \Rightarrow \sin \theta - \sqrt{3} \cos \theta = 0\]

\[ \Rightarrow \sin \theta = \sqrt{3} \cos \theta\]

\[ \Rightarrow \tan \theta = \sqrt{3} \]

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