Write the value of `hat(i).(hat(j)xxhat(k))+hat(j).(hat(i)xxhat(k))+hat(k).(hat(i)xxhat(j))`.

Explanation

To solve the expression \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \), we will use the properties of the dot product and the cross product of unit vectors.

### Step-by-Step Solution:

1. **Evaluate \( \hat{j} \times \hat{k} \)**:
\[ \hat{j} \times \hat{k} = \hat{i} \]
This is because the cross product of unit vectors follows the right-hand rule.

2. **Substitute into the expression**:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} \]
Since \( \hat{j} \times \hat{k} = \hat{i} \).

3. **Calculate \( \hat{i} \cdot \hat{i} \)**:
\[ \hat{i} \cdot \hat{i} = 1 \]

4. **Evaluate \( \hat{i} \times \hat{k} \)**:
\[ \hat{i} \times \hat{k} = -\hat{j} \]
Again, using the right-hand rule.

5. **Substitute into the expression**:
\[ \hat{j} \cdot (\hat{i} \times \hat{k}) = \hat{j} \cdot (-\hat{j}) \]

6. **Calculate \( \hat{j} \cdot (-\hat{j}) \)**:
\[ \hat{j} \cdot (-\hat{j}) = -1 \]

7. **Evaluate \( \hat{i} \times \hat{j} \)**:
\[ \hat{i} \times \hat{j} = \hat{k} \]

8. **Substitute into the expression**:
\[ \hat{k} \cdot (\hat{i} \times \hat{j}) = \hat{k} \cdot \hat{k} \]

9. **Calculate \( \hat{k} \cdot \hat{k} \)**:
\[ \hat{k} \cdot \hat{k} = 1 \]

10. **Combine all parts**:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = 1 - 1 + 1 \]

11. **Final result**:
\[ 1 - 1 + 1 = 1 \]

### Final Answer:
The value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \) is \( 1 \).