Question

If a line in octant OXYZ makes equal angles with the coordinate axes, then

• A. \( l = m = n = \frac{1}{3} \)
• B. \( l = m = n = -\frac{1}{3} \)
• C. \( l = m = n = \frac{1}{\sqrt{3}} \)
• D. \( l = m = n = -\frac{1}{\sqrt{3}} \)

Hint:

When a line makes equal angles with all axes, its direction cosines are equal in magnitude.

Answer 1

The Correct Option is C

Step 1: Recall the condition on direction cosines.
For a line with direction cosines \( l, m, n \), \[ l^2 + m^2 + n^2 = 1 \]

Step 2: Use the given condition.
Since the line makes equal angles with the coordinate axes, \[ l = m = n \]

Step 3: Substitute in the identity.
\[ 3l^2 = 1 \Rightarrow l^2 = \frac{1}{3} \Rightarrow l = \frac{1}{\sqrt{3}} \]

Step 4: Sign of direction cosines.
As the line lies in the first octant, all direction cosines are positive.

Step 5: Conclusion.
\[ \boxed{l = m = n = \frac{1}{\sqrt{3}}} \]