Question

Find the vector $\overrightarrow{a}$ of magnitude $5\sqrt{2}$, making an angle of $\frac{\pi }{4}$ with x-axis, $\frac{\pi }{2}$ with y-axis and an angle $\theta$ with z-axis.

Answer 1

$\text{Direction cosines of required vector}$$\overrightarrow{a}are$
$l=\cos \frac{\pi }{4}=\frac{1}{\sqrt{2}},m=\cos \frac{\pi }{2}=0,n=\cos \theta$

$As,l^2+m^2+n^2=1\Rightarrow \frac{1}{2}+0+n^2=1$

$\Rightarrow n^2=1-\frac{1}{2}=\frac{1}{2}\Rightarrow n=\cos \theta =\pm \frac{1}{\sqrt{2}}$

$\text{Required vector}$ $\overrightarrow{r}=5\sqrt{2}(l\hat{i}+m\hat{j}+n\hat{k})$

$=5\sqrt{2}(\frac{1}{\sqrt{2}}\hat{i}+0\hat{j}\pm \frac{1}{\sqrt{2}}\hat{k})$

$=5\hat{i}\pm 5\hat{k}$