If \( \alpha, \beta, \gamma \) are the angles which a line makes with positive directions of \( x, y, z \) axes respectively, then which of the following is not true?
If \( \alpha, \beta, \gamma \) are the angles which a line makes with positive directions of \( x, y, z \) axes respectively, then which of the following is not true?
Show Hint
- \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \)
- \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \)
- \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1 \)
- \( \cos \alpha + \cos \beta + \cos \gamma = 1 \)
The Correct Option is D
Solution and Explanation
Step 1: Recall the direction cosine property
The sum of the squares of the cosines of the angles a line makes with the coordinate axes is: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1. \] This is a fundamental property of direction cosines.
Step 2: Check each option
Option (A): True, as it is the direction cosine property.
Option (B): True, as \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = 2 \).
Option (C): True, derived from trigonometric identities for direction cosines.
Option (D): False, as \( \cos \alpha + \cos \beta + \cos \gamma \neq 1 \) in general.
Step 3: Conclude the result
Option (D) is not true.
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