If the direction cosines of a line are \( \sqrt{3}k, \sqrt{3}k, \sqrt{3}k \), then the value of \( k \) is:
Show Hint
- \( \pm 1 \)
- \( \pm \sqrt{3} \)
- \( \pm 3 \)
- \( \pm \frac{1}{3} \)
The Correct Option is D
Solution and Explanation
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) \( \pm \frac{1}{3} \).
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