If the vectors `hat(i)+hat(j)+hat(k)` makes angle `alpha, beta and gamma` with vectors `hat(i), hat(j) and hat(k)` respectively, then

Options

A. `alpha=beta ne gamma`
B. `alpha=gammanebeta`
C. `beta=gammanealpha`
D. `alpha=beta=gamma`

Answer

To solve the problem, we need to find the relationship between the angles \( \alpha \), \( \beta \), and \( \gamma \) that the vector \( \hat{i} + \hat{j} + \hat{k} \) makes with the unit vectors \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) respectively.

### Step-by-Step Solution:

1. **Define the Vector:**
Let \( \mathbf{A} = \hat{i} + \hat{j} + \hat{k} \).

2. **Calculate the Magnitude of Vector \( \mathbf{A} \):**
\[ |\mathbf{A}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]

3. **Calculate the Dot Product with \( \hat{i} \):**
The angle \( \alpha \) is defined such that:
\[ \mathbf{A} \cdot \hat{i} = |\mathbf{A}| |\hat{i}| \cos \alpha \]
Substituting the values:
\[ \hat{i} + \hat{j} + \hat{k} \cdot \hat{i} = 1 \cdot \sqrt{3} \cdot 1 \cdot \cos \alpha \]
This simplifies to:
\[ 1 = \sqrt{3} \cos \alpha \]
Thus,
\[ \cos \alpha = \frac{1}{\sqrt{3}} \]

4. **Calculate the Dot Product with \( \hat{j} \):**
Similarly, for angle \( \beta \):
\[ \mathbf{A} \cdot \hat{j} = |\mathbf{A}| |\hat{j}| \cos \beta \]
This gives:
\[ \hat{i} + \hat{j} + \hat{k} \cdot \hat{j} = 1 \cdot \sqrt{3} \cdot 1 \cdot \cos \beta \]
Simplifying:
\[ 1 = \sqrt{3} \cos \beta \]
Thus,
\[ \cos \beta = \frac{1}{\sqrt{3}} \]

5. **Calculate the Dot Product with \( \hat{k} \):**
For angle \( \gamma \):
\[ \mathbf{A} \cdot \hat{k} = |\mathbf{A}| |\hat{k}| \cos \gamma \]
This gives:
\[ \hat{i} + \hat{j} + \hat{k} \cdot \hat{k} = 1 \cdot \sqrt{3} \cdot 1 \cdot \cos \gamma \]
Simplifying:
\[ 1 = \sqrt{3} \cos \gamma \]
Thus,
\[ \cos \gamma = \frac{1}{\sqrt{3}} \]

6. **Establish the Relationship Between Angles:**
Since we have:
\[ \cos \alpha = \cos \beta = \cos \gamma = \frac{1}{\sqrt{3}} \]
This implies that:
\[ \alpha = \beta = \gamma \]

### Conclusion:
The angles \( \alpha \), \( \beta \), and \( \gamma \) are equal.

### Final Answer:
\[ \alpha = \beta = \gamma \]