Question

The number of direction cosines of vector $\overrightarrow{r}$ which is equally inclined to $OX,OY$ and $OZ$ are

Answer 1

Let $l,m,n$ be the direction cosines of $\overrightarrow{r}.$
Since $\overrightarrow{r}$ is equally inclined with $x,y$ and $z$-axis, $l=m=n$
$\therefore l^2+m^2+n^2=1$

$\Rightarrow {3l}^2=1$
$\Rightarrow l=\pm \frac{1}{\sqrt{3}}$

$\therefore$ direction cosines of $\overrightarrow{r}$ are $\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}$
Now, $\overrightarrow{r}=\left|\overrightarrow{r}\right|(l\hat{i}+m\hat{k}+n\hat{k})\Rightarrow \overrightarrow{r}=\left|\overrightarrow{r}\right|(\pm \frac{1}{\sqrt{3}}\hat{i}\pm \frac{1}{\sqrt{3}}\hat{j}\pm \frac{1}{\sqrt{3}}\hat{k})$
Since $+$ and $-$ signs can be arranged at three places, there are eight vectors, i.e.,$2\times 2\times 2$ which are equally inclined to axes.