If P is a point in space such that OP is inclined to OX at `45^(@)` and OY to `60^(@)` then OP inclined to ZO at

Options

A. `75^(@)`
B. `60^(@) and 120^(@)`
C. `75^(@) and 105^(@)`
D. `255^(@)`

Answer

To solve the problem, we need to find the angle at which the line segment OP is inclined to the Z-axis (ZO), given that it is inclined to the X-axis (OX) at 45 degrees and to the Y-axis (OY) at 60 degrees.

### Step-by-Step Solution:

1. **Understanding the Angles**:
- Let the angle between OP and OX be \( \alpha = 45^\circ \).
- Let the angle between OP and OY be \( \beta = 60^\circ \).
- Let the angle between OP and OZ be \( \gamma \) (which we need to find).

2. **Using the Cosine Rule**:
- According to the cosine rule for three-dimensional coordinates, we have:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Substituting Known Values**:
- We know:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \quad \text{and} \quad \cos 60^\circ = \frac{1}{2} \]
- Therefore, substituting these values into the equation gives:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + \cos^2 \gamma = 1 \]
- This simplifies to:
\[ \frac{1}{2} + \frac{1}{4} + \cos^2 \gamma = 1 \]

4. **Finding a Common Denominator**:
- The common denominator for \( \frac{1}{2} \) and \( \frac{1}{4} \) is 4:
\[ \frac{2}{4} + \frac{1}{4} + \cos^2 \gamma = 1 \]
- This simplifies to:
\[ \frac{3}{4} + \cos^2 \gamma = 1 \]

5. **Isolating \( \cos^2 \gamma \)**:
- Rearranging gives:
\[ \cos^2 \gamma = 1 - \frac{3}{4} = \frac{1}{4} \]

6. **Finding \( \cos \gamma \)**:
- Taking the square root:
\[ \cos \gamma = \pm \frac{1}{2} \]

7. **Finding the Angles**:
- If \( \cos \gamma = \frac{1}{2} \), then \( \gamma = 60^\circ \).
- If \( \cos \gamma = -\frac{1}{2} \), then \( \gamma = 120^\circ \).

### Conclusion:
The angle at which OP is inclined to the Z-axis (ZO) can be either \( 60^\circ \) or \( 120^\circ \).