If a vector `vecA` makes angles `45^(@)` and `60^(@)` with x and y axis respectively then the angle made by it with z-axis is

Options

A. `30^(@)`
B. `60^(@)`
C. `90^(@)`
D. `120^(@)`

Explanation

To find the angle made by the vector \(\vec{A}\) with the z-axis given that it makes angles of \(45^\circ\) with the x-axis and \(60^\circ\) with the y-axis, we can use the relationship between the angles and the cosines of these angles.

### Step-by-Step Solution:

1. **Understand the Angles**:
- Let \(\alpha\) be the angle with the x-axis, \(\beta\) be the angle with the y-axis, and \(\gamma\) be the angle with the z-axis.
- Given: \(\alpha = 45^\circ\) and \(\beta = 60^\circ\).

2. **Use the Cosine Relationship**:
- The relationship between the angles is given by the equation:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Calculate \(\cos^2 \alpha\) and \(\cos^2 \beta\)**:
- Calculate \(\cos^2 45^\circ\):
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} \implies \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
- Calculate \(\cos^2 60^\circ\):
\[ \cos 60^\circ = \frac{1}{2} \implies \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

4. **Substitute Values into the Equation**:
- Substitute \(\cos^2 \alpha\) and \(\cos^2 \beta\) into the equation:
\[ \frac{1}{2} + \frac{1}{4} + \cos^2 \gamma = 1 \]

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

6. **Find \(\cos \gamma\)**:
- Taking the square root:
\[ \cos \gamma = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

7. **Determine \(\gamma\)**:
- The angle \(\gamma\) corresponding to \(\cos \gamma = \frac{1}{2}\) is:
\[ \gamma = 60^\circ \]

### Final Answer:
The angle made by the vector \(\vec{A}\) with the z-axis is \(60^\circ\).

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