Question

The number of straight lines that are equally inclined to three-dimensional coordinate axes, is

Answer 1

If $\alpha ,\beta ,\gamma$ are the angles made by the line with $x,y,$ and $z$-axis, respectively, then

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

Given, $\alpha =\beta =\gamma$

$\therefore 3{\cos }^2\alpha =1$ or, $\cos \alpha =\pm \frac{1}{\sqrt{3}}$

Possible direction cosines are $(\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}}).$

Different sets of $d.c^{\prime }s$ are $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}),(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}),(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}})$

and $(\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$

Thus, four lines are equally inclined to axes.