Question

If $\alpha, \beta$, and $\gamma$ are the angles which a line makes with the positive directions of $x, y, z$ axes respectively, then which of the following is not true?

• A. $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$
• B. $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 1$
• C. $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1$
• D. $\cos \alpha + \cos \beta + \cos \gamma = 1$

Hint:

Direction cosines satisfy $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$, independent of their sums.

Answer 1

The Correct Option is D

For a line making angles $\alpha, \beta, \gamma$ with the coordinate axes, the equation: $$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$ is always true because it represents the property of direction cosines. The statement $\cos \alpha + \cos \beta + \cos \gamma = 1$ is not valid since it assumes specific alignment which is not general for direction cosines.
Final Answer: $\boxed{ {(D)}}$