The value of the following expression
`hati.(hatjxxhatk)+j.(hatixxhatk)+hatk.(hatjxxhati)`is a) 0 b) 1 c) -1 d) 3

Options

A. 0
B. 1
C. `-1`
D. 3

Answer

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

### Step 1: Calculate \( \hat{j} \times \hat{k} \)
Using the right-hand rule for cross products:
\[ \hat{j} \times \hat{k} = \hat{i} \]

### Step 2: Calculate \( \hat{i} \cdot (\hat{j} \times \hat{k}) \)
Now substituting the result from Step 1:
\[ \hat{i} \cdot \hat{i} = 1 \]

### Step 3: Calculate \( \hat{i} \times \hat{k} \)
Using the right-hand rule for cross products:
\[ \hat{i} \times \hat{k} = -\hat{j} \]

### Step 4: Calculate \( \hat{j} \cdot (\hat{i} \times \hat{k}) \)
Now substituting the result from Step 3:
\[ \hat{j} \cdot (-\hat{j}) = -1 \]

### Step 5: Calculate \( \hat{j} \times \hat{i} \)
Using the right-hand rule for cross products:
\[ \hat{j} \times \hat{i} = -\hat{k} \]

### Step 6: Calculate \( \hat{k} \cdot (\hat{j} \times \hat{i}) \)
Now substituting the result from Step 5:
\[ \hat{k} \cdot (-\hat{k}) = -1 \]

### Step 7: Combine all the results
Now we combine all the results from Steps 2, 4, and 6:
\[ 1 + (-1) + (-1) = 1 - 1 - 1 = -1 \]

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