`(hat(k)xx hat(j)).hat(i)+hat(j).hat(k)=` ………….

Answer

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

### Step 1: Calculate \(\hat{k} \times \hat{j}\)
Using the right-hand rule for cross products, we can determine the result of \(\hat{k} \times \hat{j}\).
- The order of the vectors is important: \(\hat{k} \times \hat{j} = -\hat{i}\).

### Step 2: Substitute into the expression
Now we substitute \(\hat{k} \times \hat{j}\) into the expression:
\[ (\hat{k} \times \hat{j}) \cdot \hat{i} = (-\hat{i}) \cdot \hat{i} \]

### Step 3: Calculate \((- \hat{i}) \cdot \hat{i}\)
The dot product of a unit vector with itself is:
\[ -\hat{i} \cdot \hat{i} = -1 \]

### Step 4: Calculate \(\hat{j} \cdot \hat{k}\)
Next, we calculate \(\hat{j} \cdot \hat{k}\):
- Since \(\hat{j}\) and \(\hat{k}\) are perpendicular, we have:
\[ \hat{j} \cdot \hat{k} = 0 \]

### Step 5: Combine the results
Now we combine the results from steps 3 and 4:
\[ -1 + 0 = -1 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{-1} \]

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