Question

If a unit vector $\overrightarrow{a}$makes an angle $\frac{\mathrm{\pi }}{3}\mathrm{with}\hat{i},\frac{\mathrm{\pi }}{4}\mathrm{with}\hat{j}$ and an acute angle θ with $\hat{k}$, then find the value of θ.

Answer 1

$$\begin{gathered} \mathrm{Since}\mathrm{a}\mathrm{unit}\mathrm{vector}\mathrm{makes}\mathrm{an}\mathrm{angle}\mathrm{of}\frac{\mathrm{\pi }}{3}\mathrm{with}\hat{\mathrm{i}},\frac{\mathrm{\pi }}{4}\mathrm{with}\hat{\mathrm{j}}\mathrm{and}\mathrm{an}\mathrm{acute}\mathrm{angle}\theta \mathrm{with}\hat{\mathrm{k}},l\mathit{}=\cos \frac{\mathrm{\pi }}{3}\mathrm{or}\frac{1}{2},m\mathit{}=\cos \frac{\mathrm{\pi }}{4}\mathrm{or}\frac{1}{\sqrt{2}}\mathrm{and}n=\cos \theta . \\ \mathrm{We}\mathrm{know} \\ {l}^2+m^{\mathit{2}}+n^{\mathit{2}}=1 \\ \Rightarrow {\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+{\cos }^2\theta =1 \\ \Rightarrow \frac{1}{4}+\frac{1}{2}+{\cos }^2\theta =1 \\ \Rightarrow {\cos }^2\theta =\frac{1}{4} \\ \Rightarrow \cos \theta =\frac{1}{2} \\ \Rightarrow \theta =\frac{\pi }{3} \\ \mathrm{Thus},\mathrm{the}\mathrm{vecto}r\mathit{}\overrightarrow{a}\mathrm{makes}\mathrm{an}\mathrm{angle}\mathrm{of}\frac{\pi }{3}\mathrm{with}\hat{k}. \end{gathered}$$