The value of `hat(i).(hat(j) xx hat(k))+ hat(j). (hat(i) xxhat(k)) + hat(k) . (hat(i) xx hat(j))`

Options

A. 0
B. -1
C. 1
D. 3

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 cross product of unit vectors.

### Step-by-Step Solution:

1. **Identify the Cross Products**:
- We know the following cross product results:
\[ \hat{j} \times \hat{k} = \hat{i} \]
\[ \hat{i} \times \hat{k} = -\hat{j} \]
\[ \hat{i} \times \hat{j} = \hat{k} \]

2. **Substitute the Cross Products into the Expression**:
- Substitute the results from step 1 into the original expression:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1 \]
\[ \hat{j} \cdot (\hat{i} \times \hat{k}) = \hat{j} \cdot (-\hat{j}) = -1 \]
\[ \hat{k} \cdot (\hat{i} \times \hat{j}) = \hat{k} \cdot \hat{k} = 1 \]

3. **Combine the Results**:
- Now, combine the results from the substitutions:
\[ 1 + (-1) + 1 = 1 - 1 + 1 = 1 \]

4. **Final Result**:
- The value of the expression is:
\[ \boxed{1} \]