The direction cosines of a line which makes equal angles with the co-ordinate axes is/are:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\pm\displaystyle\frac{1}{\sqrt{3}},\ \pm\displaystyle\frac{1}{\sqrt{3}},\ \pm\displaystyle\frac{1}{\sqrt{3}}\)
Let the line make the angle α with each axis. Then its direction cosines are
\(l = m = n = \cos\alpha\)
Direction cosines always satisfy \(l^2 + m^2 + n^2 = 1\):
\(3\cos^2\alpha = 1\)
\( \displaystyle \cos\alpha = \pm\frac{1}{\sqrt{3}} \)
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 line, read in the opposite direction).
Why the other options are incorrect
Option (1) 1, 1, 1 — Incorrect. These are direction ratios, not direction cosines: \(1^2 + 1^2 + 1^2 = 3 \ne 1\).
Option (2) −1, −1, −1 — Incorrect. Again the squares add up to 3, not 1.
Option (4) \(\pm\sqrt{3},\ \pm\sqrt{3},\ \pm\sqrt{3}\) — Incorrect. The squares add up to 9, not 1; a direction cosine can never exceed 1 in size.
NCERT (2026-27 edition):
Direction Cosines and Direction Ratios of a Line
