If the line `vec(OR)` makes angles ` theta_(1),theta_(2),theta_(3)` with the planes ` XOY, YOZ, ZOX` respectively , then ` cos^(2)theta_(1)+cos^(2)theta_(2)+cos^(2)theta_(3)` is equal to

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the value of \( \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 \) where \( \theta_1, \theta_2, \theta_3 \) are the angles that the line \( \vec{OR} \) makes with the planes \( XY \), \( YZ \), and \( ZX \) respectively.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
- The angles \( \theta_1, \theta_2, \theta_3 \) are the angles that the line makes with the coordinate planes. The cosines of these angles are known as direction cosines.
- Let \( l = \cos \theta_1 \), \( m = \cos \theta_2 \), and \( n = \cos \theta_3 \).

2. **Using the Property of Direction Cosines**:
- A fundamental property of direction cosines states that for any line in three-dimensional space, the sum of the squares of the direction cosines is equal to 1:
\[ l^2 + m^2 + n^2 = 1 \]

3. **Substituting the Direction Cosines**:
- Substituting the values of the direction cosines:
\[ \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 = l^2 + m^2 + n^2 \]

4. **Conclusion**:
- From the property of direction cosines, we know that:
\[ \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 = 1 \]
- Therefore, the final answer is:
\[ \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 = 1 \]

### Final Answer:
\[ \cos^2 \theta_1 + \cos^2 \theta_2 + \cos^2 \theta_3 = 1 \]