Question

If the direction cosines of a line are \( \sqrt{3}k, \sqrt{3}k, \sqrt{3}k \), then the value of \( k \) is:

• A. \( \pm 1 \)
• B. \( \pm \sqrt{3} \)
• C. \( \pm 3 \)
• D. \( \pm \frac{1}{3} \)
• E.

Hint:

The direction cosines of a line always satisfy the relation \( l^2 + m^2 + n^2 = 1 \). Use this property to solve for unknown parameters in problems involving direction cosines.

Answer 1

The Correct Option is D

The direction cosines of a line, denoted by \( l, m, n \), satisfy the condition: \[ l^2 + m^2 + n^2 = 1. \] Here, the direction cosines are \( l = \sqrt{3}k \), \( m = \sqrt{3}k \), and \( n = \sqrt{3}k \). Substituting these into the equation: \[ (\sqrt{3}k)^2 + (\sqrt{3}k)^2 + (\sqrt{3}k)^2 = 1. \] Simplify the terms: \[ 3k^2 + 3k^2 + 3k^2 = 1, \] \[ 9k^2 = 1. \] Solve for \( k^2 \): \[ k^2 = \frac{1}{9}. \] Take the square root on both sides: \[ k = \pm \frac{1}{3}. \] Hence, the value of \( k \) is (D) \( \pm \frac{1}{3} \).