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. \( \pm \frac{1}{3} \)

Hint:

Direction cosines of a line always satisfy the equation \( l^2 + m^2 + n^2 = 1 \). Use this to determine unknown values.

Answer 1

The Correct Option is D

The direction cosines of a line satisfy the relation: \[ l^2 + m^2 + n^2 = 1, \] 
where \( l, m, n \) are the direction cosines. 

Here: \[ l = \sqrt{3}k, \, m = \sqrt{3}k, \, n = \sqrt{3}k. \] 

Substitute into the equation: \[ (\sqrt{3}k)^2 + (\sqrt{3}k)^2 + (\sqrt{3}k)^2 = 1. \] 

Simplify: \[ 3k^2 + 3k^2 + 3k^2 = 1 \implies 9k^2 = 1 \implies k^2 = \frac{1}{9}. \] Thus: \[ k = \pm \frac{1}{3}. \] 

The correct answer is (D) \( \pm \frac{1}{3} \).