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, denoted by \( l, m, n \), adhere to the fundamental relationship: \[ l^2 + m^2 + n^2 = 1. \]
Given that \( l = \sqrt{3}k, \, m = \sqrt{3}k, \) and \( n = \sqrt{3}k \), we substitute these into the equation: \[ (\sqrt{3}k)^2 + (\sqrt{3}k)^2 + (\sqrt{3}k)^2 = 1. \] 

Simplifying the expression yields: \[ 3k^2 + 3k^2 + 3k^2 = 1, \] which further simplifies to \[ 9k^2 = 1. \] Consequently, \( k^2 = \frac{1}{9} \). 

Solving for \( k \), we find: \[ k = \pm \frac{1}{3}. \] 

The determined value for \( k \) is \( \pm \frac{1}{3} \), corresponding to option (D).