Evaluated: `hati (hatj xx hatk ) +(hatixx hatk). hatj`

Answer

To evaluate the expression \( \hat{i} (\hat{j} \times \hat{k}) + (\hat{i} \times \hat{k}) \cdot \hat{j} \), we will follow these steps:

### Step 1: Evaluate the Cross Products
First, we need to evaluate the cross products in the expression.

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

2. **Calculate \( \hat{i} \times \hat{k} \)**:
\[ \hat{i} \times \hat{k} = -\hat{j} \]

### Step 2: Substitute the Results Back into the Expression
Now, substitute the results of the cross products back into the original expression:

\[ \hat{i} (\hat{j} \times \hat{k}) + (\hat{i} \times \hat{k}) \cdot \hat{j} = \hat{i} \hat{i} + (-\hat{j}) \cdot \hat{j} \]

### Step 3: Evaluate the Dot Products
Next, we evaluate the dot products in the expression:

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

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

### Step 4: Combine the Results
Now, combine the results of the dot products:

\[ 1 + (-1) = 1 - 1 = 0 \]

### Final Result
Thus, the final result of the expression is:

\[ \boxed{0} \]