The angles with a vector `hat(i)+hat(j)+sqrt(2hat(k))` makes with X,Y and Z axes respectively are

Options

A. `60^(@),60^(@),60^(@)`
B. `45^(@), 45^(@), 45^(@)`
C. `60^(@), 60^(@),45^(@)`
D. `45^(@), 45^(@), 60^(@)`

Explanation

To find the angles that the vector \( \hat{i} + \hat{j} + \sqrt{2} \hat{k} \) makes with the X, Y, and Z axes, we can use the concept of the dot product and the magnitudes of the vectors involved.

### Step-by-Step Solution:

1. **Define the Vector**:
The given vector is
\[ \mathbf{A} = \hat{i} + \hat{j} + \sqrt{2} \hat{k} \]

2. **Find the Magnitude of the Vector**:
The magnitude of vector \( \mathbf{A} \) is calculated as follows:
\[ |\mathbf{A}| = \sqrt{(1)^2 + (1)^2 + (\sqrt{2})^2} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2 \]

3. **Calculate the Angle with the X-axis**:
Let \( \alpha \) be the angle with the X-axis. We use the dot product:
\[ \cos(\alpha) = \frac{\mathbf{A} \cdot \hat{i}}{|\mathbf{A}| |\hat{i}|} \]
Here, \( \hat{i} \) has a magnitude of 1, and the dot product \( \mathbf{A} \cdot \hat{i} = 1 \).
Thus,
\[ \cos(\alpha) = \frac{1}{2} \implies \alpha = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

4. **Calculate the Angle with the Y-axis**:
Let \( \beta \) be the angle with the Y-axis. Similarly,
\[ \cos(\beta) = \frac{\mathbf{A} \cdot \hat{j}}{|\mathbf{A}| |\hat{j}|} \]
The dot product \( \mathbf{A} \cdot \hat{j} = 1 \).
Thus,
\[ \cos(\beta) = \frac{1}{2} \implies \beta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

5. **Calculate the Angle with the Z-axis**:
Let \( \gamma \) be the angle with the Z-axis. We find:
\[ \cos(\gamma) = \frac{\mathbf{A} \cdot \hat{k}}{|\mathbf{A}| |\hat{k}|} \]
The dot product \( \mathbf{A} \cdot \hat{k} = \sqrt{2} \).
Thus,
\[ \cos(\gamma) = \frac{\sqrt{2}}{2} \implies \gamma = \cos^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45^\circ \]

6. **Final Angles**:
Therefore, the angles that the vector makes with the X, Y, and Z axes are:
\[ \alpha = 60^\circ, \quad \beta = 60^\circ, \quad \gamma = 45^\circ \]

### Summary of Results:
- Angle with X-axis (\( \alpha \)): \( 60^\circ \)
- Angle with Y-axis (\( \beta \)): \( 60^\circ \)
- Angle with Z-axis (\( \gamma \)): \( 45^\circ \)