Question

The direction cosines of a line which makes equal angles with the coordinate axes is/are
$\left(a\right)\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\left(b\right)\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}\left(c\right)-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\left(d\right)\pm \frac{1}{3},\pm \frac{1}{3},\pm \frac{1}{3}$

Answer 1

Let the line makes angle $\alpha$ with the coordinate axes.
Then, its direction cosines are $\cos \alpha ,\cos \alpha ,\cos \alpha$.

We know that,
${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
$\Rightarrow {\cos }^2\alpha +{\cos }^2\alpha +{\cos }^2\alpha =1$
$\Rightarrow 3{\cos }^2\alpha =1$
$\therefore \cos \alpha =\pm \frac{1}{\sqrt{3}}$
So, the direction cosines are $\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}$

Answer: $\left(b\right)$