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