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 forming angles \( \alpha, \beta, \gamma \) with the coordinate axes, the relation \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \) is universally applicable, reflecting the fundamental property of direction cosines. Conversely, the expression \( \cos \alpha + \cos \beta + \cos \gamma = 1 \) is not generally valid as it implies a particular orientation not characteristic of direction cosines.
Final Answer: \( \boxed{ {(D)}} \)