Step 1: Understanding the Concept:
The angles \( \alpha, \beta, \gamma \) are the direction angles of the ray. Their cosines, \( \cos \alpha, \cos \beta, \cos \gamma \), are called direction cosines (typically denoted as \( l, m, n \)).
: Key Formula or Approach:
Identity for direction cosines: \( \cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1 \).
Step 2: Detailed Explanation:
We know that:
\[ \cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1 \]
Using the trigonometric identity \( \cos^{2} \theta = 1 - \sin^{2} \theta \), substitute for all three terms:
\[ (1 - \sin^{2} \alpha) + (1 - \sin^{2} \beta) + (1 - \sin^{2} \gamma) = 1 \]
\[ 1 + 1 + 1 - (\sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma) = 1 \]
\[ 3 - (\sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma) = 1 \]
Rearranging to find the sum:
\[ \sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma = 3 - 1 \]
\[ \sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma = 2 \].
Step 3: Final Answer:
The value is 2.