If a line makes angle `90^(@)` and `60^(@)` respectively with positively direction of x and y axes, find the angle which it makes with the positive direction of z -axis.

Answer

To find the angle that a line makes with the positive direction of the z-axis, given that it makes angles of \(90^\circ\) and \(60^\circ\) with the positive directions of the x-axis and y-axis respectively, we can use the relationship between the direction cosines of the angles.

### Step-by-Step Solution:

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

From the problem, we have:
\[ \alpha = 90^\circ, \quad \beta = 60^\circ \]

2. **Use the Direction Cosine Formula**:
The relationship between the direction cosines is given by:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Calculate \(\cos \alpha\) and \(\cos \beta\)**:
- Since \(\alpha = 90^\circ\):
\[ \cos \alpha = \cos 90^\circ = 0 \]
- For \(\beta = 60^\circ\):
\[ \cos \beta = \cos 60^\circ = \frac{1}{2} \]

4. **Substitute Values into the Formula**:
Substitute the values of \(\cos \alpha\) and \(\cos \beta\) into the direction cosine formula:
\[ 0^2 + \left(\frac{1}{2}\right)^2 + \cos^2 \gamma = 1 \]
This simplifies to:
\[ 0 + \frac{1}{4} + \cos^2 \gamma = 1 \]

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

6. **Find \(\cos \gamma\)**:
Taking the square root, we find:
\[ \cos \gamma = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \quad \text{or} \quad \cos \gamma = -\frac{\sqrt{3}}{2} \]

7. **Determine the Angles**:
- If \(\cos \gamma = \frac{\sqrt{3}}{2}\), then:
\[ \gamma = 30^\circ \]
- If \(\cos \gamma = -\frac{\sqrt{3}}{2}\), then:
\[ \gamma = 150^\circ \]

### Final Answer:
The angle which the line makes with the positive direction of the z-axis can be either \(30^\circ\) or \(150^\circ\).

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