Question

A line makes equal angles with the coordinate axis. The direction cosines of this line are

• A. $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$
• B. $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$
• C. $(\frac{1}{\sqrt{3}},\frac{1}{3},\frac{1}{3})$
• D. $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$

Answer 1

The correct option is B $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$

Let the direction cosines of the line make an angle α with each of the coordinate axes.

$\therefore l=cos\alpha ,m=cos\alpha ,n=cos\alpha$

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

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

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

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

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

Thus,the direction cosines of the line,which is equally inclined to the coordinate axes,are

$\pm \frac{1}{\sqrt{3}},\pm \frac{1}{\sqrt{3}},and\pm \frac{1}{\sqrt{3}}.$