Question

What are the direction cosines of a line which is equally inclined to the positive directions of the axes?

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

Answer 1

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

We know sum of the squares of the direction cosines is one.

$i.e.co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$

but its given that $\alpha =\beta =\gamma$

$\therefore co{s}^2\alpha +co{s}^2\alpha +co{s}^2\alpha =1$

$3co{s}^2\alpha =1$

$\therefore co{s}^2\alpha =\frac{1}{3}$

$\therefore$ Positive directions of the axes are $(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$