Question

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

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is B $4$

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 $\alpha =\beta =\gamma$

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

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

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

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

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

Thus, four lines are equally inclined to axes.