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

Answer

To solve the expression \((\hat{k} \times \hat{i}) \cdot \hat{j} + \hat{i} \cdot \hat{k}\), we will break it down step by step.

### Step 1: Calculate \(\hat{k} \times \hat{i}\)

Using the right-hand rule for cross products, we know:
\[ \hat{k} \times \hat{i} = \hat{j} \]
This is because when you point your fingers in the direction of \(\hat{k}\) (upwards) and curl them towards \(\hat{i}\) (to the right), your thumb points in the direction of \(\hat{j}\) (outwards).

### Step 2: Substitute into the expression

Now we can substitute this result back into the original expression:
\[ (\hat{k} \times \hat{i}) \cdot \hat{j} = \hat{j} \cdot \hat{j} \]

### Step 3: Calculate \(\hat{j} \cdot \hat{j}\)

The dot product of a unit vector with itself is equal to 1:
\[ \hat{j} \cdot \hat{j} = 1 \]

### Step 4: Calculate \(\hat{i} \cdot \hat{k}\)

Next, we compute the dot product \(\hat{i} \cdot \hat{k}\). Since \(\hat{i}\) and \(\hat{k}\) are perpendicular to each other:
\[ \hat{i} \cdot \hat{k} = 0 \]

### Step 5: Combine the results

Now we can combine the results from Steps 3 and 4:
\[ (\hat{k} \times \hat{i}) \cdot \hat{j} + \hat{i} \cdot \hat{k} = 1 + 0 = 1 \]

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