If a line makes an angle of `pi/4` with the positive directions of each of x-axis and y-axis, then the angle that the line makes with the positive direction of z-axis is

Options

A. `pi/6`
B. `pi/4`
C. `pi/3`
D. `pi/2`

Answer

To solve the problem, we need to find the angle that a line makes with the positive direction of the z-axis, given that it makes an angle of \( \frac{\pi}{4} \) with both the positive directions of the x-axis and y-axis.

### Step-by-Step Solution:

1. **Understanding the Angles**:
Let:
- \( \alpha \) be the angle with the x-axis,
- \( \beta \) be the angle with the y-axis,
- \( \gamma \) be the angle with the z-axis.

According to the problem, we have:
\[ \alpha = \beta = \frac{\pi}{4} \]

2. **Using the Relation of Direction Cosines**:
The relationship between the direction cosines is given by:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Substituting the Known Values**:
Substitute \( \alpha \) and \( \beta \) into the equation:
\[ \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \left(\frac{\pi}{4}\right) + \cos^2 \gamma = 1 \]

4. **Calculating \( \cos^2 \left(\frac{\pi}{4}\right) \)**:
We know that:
\[ \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Therefore:
\[ \cos^2 \left(\frac{\pi}{4}\right) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

5. **Substituting Back into the Equation**:
Now substitute \( \cos^2 \left(\frac{\pi}{4}\right) \) into the equation:
\[ \frac{1}{2} + \frac{1}{2} + \cos^2 \gamma = 1 \]

6. **Simplifying the Equation**:
This simplifies to:
\[ 1 + \cos^2 \gamma = 1 \]

7. **Solving for \( \cos^2 \gamma \)**:
Subtract 1 from both sides:
\[ \cos^2 \gamma = 0 \]

8. **Finding \( \gamma \)**:
Taking the square root gives:
\[ \cos \gamma = 0 \]
This implies:
\[ \gamma = \frac{\pi}{2} \]

### Conclusion:
The angle that the line makes with the positive direction of the z-axis is \( \frac{\pi}{2} \).