Question

If the direction cosines of a line are $\lambda, \lambda, \lambda$, then $\lambda$ is equal to:

• A. $\frac{-1}{\sqrt{3}}$
• B. $1$
• C. $\frac{1}{\sqrt{3}}$
• D. $\pm \frac{1}{\sqrt{3}}$

Hint:

For direction cosines of a line, always remember that the sum of the squares of the direction cosines equals 1. Use this relation to solve for $\lambda$.

Answer 1

The Correct Option is C

We are given that the direction cosines of a line are $\lambda, \lambda, \lambda$. The sum of the squares of the direction cosines of a line is always equal to 1: \[ \lambda^2 + \lambda^2 + \lambda^2 = 1. \] This simplifies to: \[ 3\lambda^2 = 1. \] Solving for $\lambda$, we get: \[ \lambda^2 = \frac{1}{3}. \] Taking the square root of both sides, we get: \[ \lambda = \pm \frac{1}{\sqrt{3}}. \] Thus, the correct value of $\lambda$ is $\pm \frac{1}{\sqrt{3}}$.