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

Question:

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

Solution:

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

∴ l = cos αm = cos αn = cos α

$l^{2}+m^{2}+n^{2}=1$

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

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

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

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

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 $\pm \frac{1}{\sqrt{3}}$.