Question

Find the direction cosines of a line which makes equal angles with the coordinate axes.

Answer 1

Let the direction cosines of the line make an angle $a$ with each of the coordinate axes.

$\therefore$ $l=\cos a,m=\cos a,n=\cos a$

We know $l^2+m^2+n^2=1$

$\Rightarrow$ ${\cos }^2\alpha +{\cos }^2\alpha +{\cos }^2\alpha =1$

$\Rightarrow$ $3{\cos }^2\alpha =1$

$\Rightarrow$ ${\cos }^2\alpha =\frac{1}{3}$

$\Rightarrow$ $\cos \alpha =\pm \frac{1}{\sqrt{3}}$

Thus, the direction cosines of the line which is equally inclined to the coordinate axes are $\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}$ and $\pm \frac{1}{\sqrt{3}}$.