Question

Find the equation of the line passing through the origin and making equal angles with all the three co-ordinate axes.

Answer 1

Let the line makes $\alpha ,\beta ,r$ angles with the coordinates axes, (in this case, $\alpha =\beta =r)$

$\because {\cos }^2\alpha +{\cos }^2\beta +{\cos }^{2}r=1$

$\therefore 3{\cos }^2\alpha =1$ $(\because \alpha =\beta =r)$

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

$\therefore$ Direction cosine is $(\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}})$

$\therefore$ equation of required line is

$\frac{x-0}{\pm \frac{1}{\sqrt{3}}}=\frac{y-0}{\pm \frac{1}{\sqrt{3}}}=\frac{z-0}{\pm \frac{1}{\sqrt{3}}}$