The direction cosines of a line equally inclined with the co-ordinate…

The correct answer is Option (3) → \(\left(\pm\displaystyle\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}},\pm\frac{1}{\sqrt{3}}\right)\)

Let the line make the same angle \(\alpha\) with each axis. Then its direction cosines are equal:

    \(l = m = n = \cos\alpha\)

Direction cosines always satisfy \(l^2 + m^2 + n^2 = 1\):

    \(3l^2 = 1\)

    \( \displaystyle l = \pm\frac{1}{\sqrt{3}} \)

So the direction cosines are \(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\) or \(-\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}}\) (the same sign in all three). The two signs give the two opposite directions along the line.

Why the other options are incorrect

• Option (1) \((1,1,1)\) — Incorrect. These are direction ratios of the line, not direction cosines: \(1^2 + 1^2 + 1^2 = 3 \ne 1\).

• Option (2) \(\left(\displaystyle\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)\) — Incorrect. \(\frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{1}{3} \ne 1\).

• Option (4) \(\left(\displaystyle\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)\) — Incorrect. \(\frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2} \ne 1\).

NCERT (2026-27 edition):
Direction Cosines and Direction Ratios of a Line