Question

A unit vector $\overrightarrow{r}$ makes angles $\frac{\mathrm{\pi }}{3}$ and $\frac{\mathrm{\pi }}{2}$ with $\hat{j}\mathrm{and}\hat{k}$ respectively and an acute angle θ with $\hat{i}$. Find θ.

Answer 1

A unit vector makes an angle $\frac{\mathrm{\pi }}{3}$ and $\frac{\mathrm{\pi }}{2}$ with $\hat{j}$ and $\hat{k}$
Let ${\rm I},m,n$ be its direction cosines
∴ $l=\cos \theta ,m=\cos \left(\frac{\mathrm{\pi }}{3}\right)=\frac{1}{2},n=\cos \left(\frac{\mathrm{\pi }}{2}\right)=0$
Now
∴ $l^2+m^2+n^2=1$
$\Rightarrow$ $l^2+\frac{1}{4}+0=1$
$\Rightarrow$ $l^2=1-\frac{1}{4}=\frac{3}{4}$
$\Rightarrow$ $l=\pm \frac{\sqrt{3}}{2}$
∴ $\overrightarrow{r}$ makes an angle $30°,150°$ with $\stackrel{⏜}{i}$
Since, angle $\theta$ is acute.
∴ $\theta =30°$