The value of `hati.(hatj xx hatk) + hatj.(hati xx hatk) + hatk.( hati xx hatj)` is

Options

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

Answer

To solve the expression \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \), we will use the properties of the dot and cross products of unit vectors in 3D space.

### Step-by-Step Solution:

1. **Evaluate \( \hat{j} \times \hat{k} \)**:
\[ \hat{j} \times \hat{k} = \hat{i} \]
This is because the right-hand rule gives us \( \hat{i} \) when we cross \( \hat{j} \) and \( \hat{k} \).

**Hint**: Remember the cyclic nature of the unit vectors: \( \hat{i} \times \hat{j} = \hat{k} \), \( \hat{j} \times \hat{k} = \hat{i} \), and \( \hat{k} \times \hat{i} = \hat{j} \).

2. **Substitute into the expression**:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) = \hat{i} \cdot \hat{i} = 1 \]

3. **Evaluate \( \hat{i} \times \hat{k} \)**:
\[ \hat{i} \times \hat{k} = -\hat{j} \]
Again, using the right-hand rule, crossing \( \hat{i} \) and \( \hat{k} \) gives us \( -\hat{j} \).

**Hint**: The negative sign indicates the direction of the resulting vector is opposite to what is expected from the right-hand rule.

4. **Substitute into the expression**:
\[ \hat{j} \cdot (\hat{i} \times \hat{k}) = \hat{j} \cdot (-\hat{j}) = -1 \]

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

**Hint**: Again, use the cyclic nature of the unit vectors.

6. **Substitute into the expression**:
\[ \hat{k} \cdot (\hat{i} \times \hat{j}) = \hat{k} \cdot \hat{k} = 1 \]

7. **Combine all parts of the expression**:
\[ \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = 1 - 1 + 1 = 1 \]

### Final Result:
The value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \) is \( 1 \).