If the directions cosines of a line are k, k, k, then

Concept:

1) If a directed line L passing through the origin makes angles α, β, and γ with x, y, and z-axes, respectively, called direction angles, then the cosine of these angles, namely, cosα, cosβ, and cosγ are called direction cosines of the directed line L.

2) These unique direction cosines are denoted by l, m, and n, given by l = cosα, m =cosβ, and n = cosγ.

3) If (l, m, n) represents the direction cosines of a line, then l² + m² + n² = 1.

Calculation:

Given: The directions cosines of a line are k, k, k.

∴ k² + k² + k² = 1

⇒ 3k² = 1

⇒ k² = \(\frac{1}{3}\)

⇒ k = ± \(\frac{1}{\sqrt3}\)

∴ k = \(\rm \frac{1}{\sqrt{3}}\) or \(-\frac{1}{\sqrt{3}}\)

The correct answer is option 4.