Question

Number of lines is space which are equally inclined to three co-ordinate axes are?

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (A)

$2$

Let the direction cosines of the line which is equally inclined to the three coordinates axes are $l,m,$ and $n$

And, the angle subtended with each axes be $\theta$

Then,

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

$\begin{array}{c}{\cos }^2\theta +{\cos }^2\theta +{\cos }^2\theta =1\\ \\ 3{\cos }^2\theta =1\\ \\ {\cos }^2\theta =\frac{1}{3}\\ \\ \cos \theta =\pm \frac{1}{\sqrt{3}}\end{array}$

If $\cos \theta =\frac{1}{\sqrt{3}}$

then,

$l=m=n=\frac{1}{\sqrt{3}}$

If $\cos \theta =-\frac{1}{\sqrt{3}}$

then,

$l=m=n=-\frac{1}{\sqrt{3}}$

Therefore,

Two such lines are possible

Hence, option $A$ is correct answer.