`[[hat(i), hat(j), hat(k)]]+[[hat(k), hat(j), hat(i)]]+[[hat(j), hatk, hat(i)]]=`

Options

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

Answer

To solve the problem, we need to evaluate the expression:

\[ \text{Triple product of } \hat{i}, \hat{j}, \hat{k} + \text{Triple product of } \hat{k}, \hat{j}, \hat{i} + \text{Triple product of } \hat{j}, \hat{k}, \hat{i} \]

### Step 1: Understand the Triple Product
The triple product of three vectors \(\hat{a}, \hat{b}, \hat{c}\) can be expressed as:
\[ \hat{a} \cdot (\hat{b} \times \hat{c}) \]

### Step 2: Calculate Each Triple Product
1. **First Triple Product**:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) \]
Using the right-hand rule, \(\hat{j} \times \hat{k} = \hat{i}\). Therefore:
\[ \hat{i} \cdot \hat{i} = 1 \]

2. **Second Triple Product**:
\[ \hat{k} \cdot (\hat{j} \times \hat{i}) \]
Again using the right-hand rule, \(\hat{j} \times \hat{i} = -\hat{k}\). Thus:
\[ \hat{k} \cdot (-\hat{k}) = -1 \]

3. **Third Triple Product**:
\[ \hat{j} \cdot (\hat{k} \times \hat{i}) \]
Using the right-hand rule, \(\hat{k} \times \hat{i} = \hat{j}\). Therefore:
\[ \hat{j} \cdot \hat{j} = 1 \]

### Step 3: Combine the Results
Now, we can add the results of the three triple products:
\[ 1 + (-1) + 1 = 1 \]

### Final Answer
Thus, the final result of the expression is:
\[ \boxed{1} \]