The value of `(hat(k) xx hat(j)). hat(i) + hat(j).hat(k)` =

Options

A. `-1`
B. 1
C. 0
D. 2

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 in vector algebra, we know that:
\[ \hat{k} \times \hat{j} = -\hat{i} \]

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

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

### Step 4: Calculate \(\hat{j} \cdot \hat{k}\)
Since \(\hat{j}\) and \(\hat{k}\) are orthogonal unit vectors, their dot product is:
\[ \hat{j} \cdot \hat{k} = 0 \]

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

### Final Result
Thus, the value of the expression \((\hat{k} \times \hat{j}) \cdot \hat{i} + \hat{j} \cdot \hat{k}\) is:
\[ \boxed{-1} \]