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

The direction cosines of a line are given as $\lambda, \lambda, \lambda$. The sum of the squares of the direction cosines of any line is invariably 1. Therefore:\[\lambda^2 + \lambda^2 + \lambda^2 = 1.\]This equation reduces to:\[3\lambda^2 = 1.\]Solving for $\lambda^2$ yields:\[\lambda^2 = \frac{1}{3}.\]Taking the square root of both sides gives:\[\lambda = \pm \frac{1}{\sqrt{3}}.\]Consequently, the value of $\lambda$ is $\pm \frac{1}{\sqrt{3}}$.