Question

The number of straight lines that are equally inclined to the three-dimensional co-ordinate axis, is

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is B

$4$

Explanation for the correct option:

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

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

we have given $\alpha =\beta =\gamma$

$$\begin{gathered} \therefore 3{\cos }^2\alpha =1 \\ \Rightarrow \cos \alpha =\pm \frac{1}{\sqrt{3}} \end{gathered}$$

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

Different direction cosines are $\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right),\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}\right),\left(\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$and $\left(\frac{-1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)$.

Thus, $4$ lines are equally inclined to axes.

Hence, option (B) is the correct answer.