Question

The direction cosines of a line which makes equal acute angles with the co-ordinate axes 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. \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \)
• D. \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \)

Hint:

For a line making equal acute angles with the coordinate axes, the direction cosines are all equal in magnitude and satisfy \( l^2 + m^2 + n^2 = 1 \).

Answer 1

The Correct Option is C

Step 1: Equal acute angles condition.
For a line making equal acute angles with the coordinate axes, the direction cosines of the line must be equal in magnitude. Since the sum of squares of direction cosines is 1, we have: \[ l^2 + m^2 + n^2 = 1 \] where \( l = m = n \). Solving for \( l, m, n \), we get: \[ l = m = n = \frac{1}{\sqrt{3}} \]
Step 2: Conclusion.
Thus, the direction cosines are \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \), corresponding to option (C).