Question

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

Answer 1

The Direction cosines of vector $\overrightarrow{a}$ are $l=\cos \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{2},m=\cos \left(\frac{\mathrm{\pi }}{4}\right)=\frac{1}{\sqrt{2}},n=\cos \theta$

Therefore,
$$\begin{gathered} l^2+m^2+n^2=1 \\ \Rightarrow \frac{1}{4}+\frac{1}{2}+n^2=1 \\ \Rightarrow n^2=1-\frac{3}{4} \\ \Rightarrow n^2=\frac{1}{4} \\ \Rightarrow n=\frac{1}{2}\left[\because \overrightarrow{a}\mathrm{makes}\mathrm{acute}\mathrm{angle}\mathrm{with}\hat{k}\right] \end{gathered}$$
$$\begin{gathered} \Rightarrow \cos \theta =\frac{1}{2} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$
Since, $\overrightarrow{a}$ is the unit vector.
$$\begin{gathered} \therefore \overrightarrow{a}=l\hat{i}+m\hat{j}+n\hat{k} \\ \Rightarrow \overrightarrow{a}=\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{2}\hat{k} \end{gathered}$$
Hence, components of $\overrightarrow{a}$ are $\frac{1}{2}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{2}\hat{k}$