Question

Find the value of θ ∈(0, π/2) for which vectors $\overrightarrow{a}=\left(\sin \mathrm{\theta }\right)\hat{i}+\left(\cos \mathrm{\theta }\right)\hat{j}\mathrm{and}\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}$ are perpendicular.

Answer 1

$$\begin{gathered} \text{We have} \\ \overrightarrow{a}=\left(\sin \theta \right)\hat{i}+\left(\cos \theta \right)\hat{j} \\ \mathrm{and} \\ \overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k} \\ \text{It is given that the vectors are perpendicular.} \\ \Rightarrow \overrightarrow{a}.\overrightarrow{b}=0 \\ \Rightarrow \sin \theta -\sqrt{3}\cos \theta =0 \\ \Rightarrow \sin \theta =\sqrt{3}\cos \theta \\ \Rightarrow \tan \theta =\sqrt{3} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$