Evaluate : ` [2hati " " hatj " " hatk ] +[hati hatk hatj ] +[hatk hatj 2hati ]`

Answer

To evaluate the expression \([2\hat{i} \hat{j} \hat{k}] + [\hat{i} \hat{k} \hat{j}] + [\hat{k} \hat{j} 2\hat{i}]\), we will follow the properties of Scalar Triple Product (STP).

### Step-by-Step Solution:

1. **Identify the Scalar Triple Products**:
- The first term is \(2[\hat{i} \hat{j} \hat{k}]\).
- The second term is \([\hat{i} \hat{k} \hat{j}]\).
- The third term is \([\hat{k} \hat{j} 2\hat{i}]\).

2. **Factor out constants**:
- For the first term, we can factor out the constant \(2\):
\[ 2[\hat{i} \hat{j} \hat{k}] \]
- The second term remains as is:
\[ [\hat{i} \hat{k} \hat{j}] \]
- For the third term, we can factor out the constant \(2\):
\[ 2[\hat{k} \hat{j} \hat{i}] \]

3. **Evaluate each Scalar Triple Product**:
- The value of \([\hat{i} \hat{j} \hat{k}]\) is \(1\).
- For \([\hat{i} \hat{k} \hat{j}]\), we have one swap (from \(\hat{j}\) to \(\hat{k}\)), which gives:
\[ [\hat{i} \hat{k} \hat{j}] = -[\hat{i} \hat{j} \hat{k}] = -1 \]
- For \([\hat{k} \hat{j} \hat{i}]\), we have two swaps (from \(\hat{i}\) to \(\hat{k}\) and \(\hat{j}\) to \(\hat{i}\)), which gives:
\[ [\hat{k} \hat{j} \hat{i}] = -[\hat{i} \hat{j} \hat{k}] = -1 \]

4. **Combine the results**:
- Now substituting back into the expression:
\[ 2(1) + (-1) + 2(-1) \]
- This simplifies to:
\[ 2 - 1 - 2 = -1 \]

5. **Final Answer**:
- Therefore, the evaluated result is:
\[ \boxed{-1} \]