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:

The direction cosines \( l, m, n \) must always satisfy the equation \( l^2 + m^2 + n^2 = 1 \).

Answer 1

The Correct Option is D

Step 1: Use the property of direction cosines. For a line in 3D space, the direction cosines \( l, m, n \) satisfy the equation: \[ l^2 + m^2 + n^2 = 1 \] Step 2: Substitute the given values. Given \( l = \sqrt{3}k \), \( m = \sqrt{3}k \), \( n = \sqrt{3}k \), the equation becomes: \[ (\sqrt{3}k)^2 + (\sqrt{3}k)^2 + (\sqrt{3}k)^2 = 1 \] Step 3: Simplify the equation. \[ 3k^2 + 3k^2 + 3k^2 = 1 \] \[ 9k^2 = 1 \] \[ k^2 = \frac{1}{9} \] Step 4: Find the value of \( k \). \[ k = \pm \frac{1}{3} \] Final Answer: \[ \boxed{\pm \frac{1}{3}} \]