Question

Find the direction cosines of vector $\overrightarrow{r}$ which is equally inclined to $OX,OY$ and $OZ$. Find total number of such vectors.

• A. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}};6$
• B. $\frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}};8$
• C. $\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}};8$
• D. None of these

Answer 1

The correct option is C

$\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}};8$

Let $l,m,n$ be the direction cosines of $\overrightarrow{r}$.

Since $\overrightarrow{r}$ is equally inclined with $x,y$ and $z$ axis, $l=m=n$

$\therefore l^2+m^2+n^2\Rightarrow 3l^3=1\Rightarrow l=\pm \frac{1}{\sqrt{3}}$

$\therefore$ direction cosines of $\overrightarrow{r}$ are $\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}$

Now, $\overrightarrow{r}=\left|\overrightarrow{r}\right|(li+mj+nk)=\overrightarrow{r}=\left|\overrightarrow{r}\right|(\pm \frac{1}{\sqrt{3}}i\pm \frac{1}{\sqrt{3}}j\pm \frac{1}{\sqrt{3}}k)$

Since $+$ and $-$ signs can be arranged at three places,

$\Rightarrow$ there are eight vectors, i.e $2\times 2\times 2$ which are equally inclined to axes.