Question

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

Answer 1

Let unit vector makes angle $\alpha ,\beta$ and $\gamma$ with $\hat{i},\hat{j}$ and $\hat{k}$ respectively, then
$\alpha =\frac{\pi }{3},\beta =\frac{\pi }{4}$ and $\gamma =\theta$ (Given)
$\therefore co{s}^2\frac{\pi }{3}+co{s}^2\frac{\pi }{4}+co{s}^2\theta =1\Rightarrow {\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+co{s}^2\theta =1\Rightarrow \frac{1}{4}+\frac{1}{2}+co{s}^2\theta =1\Rightarrow co{s}^2\theta =1-\frac{3}{4}\Rightarrow co{s}^2\theta =\frac{4-3}{4}=\frac{1}{4}\Rightarrow cos\theta =\pm \frac{1}{\sqrt{4}}\Rightarrow cos\theta =\pm \frac{1}{2}$
$cos\theta =\frac{1}{2}$ $[cos\theta \ne -\frac{1}{2},\because \theta isanacuteangle]$
$\Rightarrow \theta =co{s}^{-1}\left(\frac{1}{2}\right)=co{s}^{-1}\left(cos\frac{\pi }{3}\right)$
$\Rightarrow \theta =\frac{\pi }{3}$ and components of a are $cos\frac{\pi }{3},cos\frac{\pi }{4},cos\frac{\pi }{3}$
$\Rightarrow \frac{1}{2},\frac{1}{\sqrt{2}},\frac{1}{2}$
Note For acute angle, $cos\theta >0$ and for obtuse angle $cos\theta <0$