`hat(i)*(hat(j)timeshat(k))+hat(j)*(hat(k)timeshat(i))+hat(k)*(hat(i)timeshat(j))=`

Options

A. `0`
B. `1`
C. `2`
D. `3`

Answer

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

### 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 \( \hat{j} \) and \( \hat{k} \) follows the right-hand rule.

**Hint**: Remember the right-hand rule for cross products: \( \hat{i} \times \hat{j} = \hat{k} \), \( \hat{j} \times \hat{k} = \hat{i} \), and \( \hat{k} \times \hat{i} = \hat{j} \).

2. **Substitute into the expression**:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1 \]

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

**Hint**: Use the cyclic nature of the unit vectors to determine the result of cross products.

4. **Substitute into the expression**:
\[ \hat{j} \cdot (\hat{k} \times \hat{i}) = \hat{j} \cdot \hat{j} = 1 \]

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

**Hint**: The order of the vectors in a cross product matters; reversing them changes the sign.

6. **Substitute into the expression**:
\[ \hat{k} \cdot (\hat{i} \times \hat{j}) = \hat{k} \cdot \hat{k} = 1 \]

7. **Combine all parts of the expression**:
\[ 1 + 1 + 1 = 3 \]

### Final Answer:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = 3 \]