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 with direction angles \( \alpha, \beta, \gamma \) relative to the coordinate axes, the identity \[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\] holds true, as it defines a fundamental property of direction cosines. The expression \( \cos \alpha + \cos \beta + \cos \gamma = 1 \) is incorrect because it implies a particular orientation of the line, which is not a general characteristic of direction cosines.
Final Answer: \( \boxed{{(D)}} \)