Question

The direction cosines of a line which makes equal angles with the co-ordinate axes is/are :

• A. 1, 1, 1
• B. -1, -1, -1
• C. \(±\frac{1}{\sqrt3},±\frac{1}{\sqrt3},±\frac{1}{\sqrt3}\)
• D. \(±\sqrt3,±\sqrt3,±\sqrt3\)

Answer 1

The Correct Option is C

The problem involves calculating the direction cosines of a line that makes equal angles with the coordinate axes. The direction cosines are the cosines of the angles that the line makes with the positive directions of the x, y, and z axes, respectively.

For a line making equal angles with the coordinate axes, let the angles with the x, y, and z axes be α, β, and γ, respectively. Since these angles are equal, we have:

\( \alpha = \beta = \gamma = θ \)

The direction cosines, denoted as l, m, and n, are given by:

\( l = \cos(θ) \), \( m = \cos(θ) \), \( n = \cos(θ) \)

Since the line makes equal angles with the axes, the sum of the squares of the direction cosines is:

\( l^2 + m^2 + n^2 = 1 \)

Substituting \( l = m = n = \cos(θ) \) into the equation:

\( 3\cos^2(θ) = 1 \)

Simplifying, we get:

\( \cos^2(θ) = \frac{1}{3} \)

Taking the square root on both sides gives:

\( \cos(θ) = ±\frac{1}{\sqrt{3}} \)

Therefore, the direction cosines are:

\( l = ±\frac{1}{\sqrt{3}}, m = ±\frac{1}{\sqrt{3}}, n = ±\frac{1}{\sqrt{3}} \)

The correct answer is: \(±\frac{1}{\sqrt3},±\frac{1}{\sqrt3},±\frac{1}{\sqrt3}\)