A line makes angles of 120°, 135° and 60° with the positive directions of the x-axis, y-axis and z-axis respectively. The direction cosines of the line are
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle -\frac{1}{2}, -\frac{1}{\sqrt{2}}, \frac{1}{2}\)
If a line makes angles \(\alpha, \beta, \gamma\) with the positive axes, its direction cosines are \(\cos\alpha, \cos\beta, \cos\gamma\).
\( \displaystyle l = \cos 120^\circ = -\frac{1}{2} \)
\( \displaystyle m = \cos 135^\circ = -\frac{1}{\sqrt{2}} \)
\( \displaystyle n = \cos 60^\circ = \frac{1}{2} \)
Check: \(\frac{1}{4}+\frac{1}{2}+\frac{1}{4} = 1\), as it must be.
Why the other options are incorrect
Option (1) \(\displaystyle\frac{1}{2}, -\frac{1}{\sqrt{2}}, \frac{1}{2}\) — Incorrect. \(\cos 120^\circ\) is negative; \(\frac{1}{2}\) is \(\cos 60^\circ\).
Option (3) \(\displaystyle -\frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2}\) — Incorrect. \(\cos 135^\circ\) is negative; \(\frac{1}{\sqrt{2}}\) is \(\cos 45^\circ\).
Option (4) \(\displaystyle -\frac{\sqrt{3}}{2}, -\frac{1}{\sqrt{2}}, \frac{1}{2}\) — Incorrect. \(-\frac{\sqrt{3}}{2}\) is \(\cos 150^\circ\), not \(\cos 120^\circ\); also the squares add to \(\frac{3}{2}\), not 1.