If `hat(i), hat(j), hat(k)` are unit vectors along three mutually perpendicular directions, then

Options

A. `hat(i).hat(j)=1`
B. `hat(i)xxhat(j) =1`
C. `hat(i).hat(k)=0`
D. `hat(i) xx hat(k)=0`

Answer

To solve the problem involving the unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) which are mutually perpendicular, we will analyze each option given in the question step by step.

### Step-by-Step Solution:

1. **Understanding the Unit Vectors**:
- The unit vectors \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) represent the x, y, and z directions respectively.
- Since they are mutually perpendicular, the angle between any two of these vectors is \(90^\circ\).

2. **Magnitude of Unit Vectors**:
- The magnitude of each unit vector is \(1\):
\[ |\hat{i}| = |\hat{j}| = |\hat{k}| = 1 \]

3. **Dot Product Calculation**:
- The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \]
- For \(\hat{i} \cdot \hat{j}\):
\[ \hat{i} \cdot \hat{j} = 1 \cdot 1 \cdot \cos(90^\circ) = 1 \cdot 1 \cdot 0 = 0 \]
- Therefore, if the option states \(\hat{i} \cdot \hat{j} = 1\), it is incorrect.

4. **Cross Product Calculation**:
- The cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is given by:
\[ \mathbf{a} \times \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \sin(\theta) \hat{n} \]
- For \(\hat{i} \times \hat{j}\):
\[ \hat{i} \times \hat{j} = 1 \cdot 1 \cdot \sin(90^\circ) \hat{k} = 1 \cdot 1 \cdot 1 \hat{k} = \hat{k} \]
- If the option states \(\hat{i} \times \hat{j} = \hat{k}\), it is correct.

5. **Checking Other Dot Products**:
- For \(\hat{i} \cdot \hat{k}\):
\[ \hat{i} \cdot \hat{k} = 1 \cdot 1 \cdot \cos(90^\circ) = 0 \]
- If the option states \(\hat{i} \cdot \hat{k} = 0\), it is correct.

6. **Checking Other Cross Products**:
- For \(\hat{i} \times \hat{k}\):
\[ \hat{i} \times \hat{k} = 1 \cdot 1 \cdot \sin(90^\circ) \hat{j} = \hat{j} \]
- If the option states \(\hat{i} \times \hat{k} = 0\), it is incorrect.

### Conclusion:
- The correct results from the calculations indicate that:
- \(\hat{i} \cdot \hat{j} = 0\) (incorrect option if stated as 1)
- \(\hat{i} \times \hat{j} = \hat{k}\) (correct)
- \(\hat{i} \cdot \hat{k} = 0\) (correct)
- \(\hat{i} \times \hat{k} = \hat{j}\) (incorrect if stated as 0)

### Final Answer:
The correct options are those that state:
- \(\hat{i} \cdot \hat{j} = 0\)
- \(\hat{i} \times \hat{j} = \hat{k}\)
- \(\hat{i} \cdot \hat{k} = 0\)
- \(\hat{i} \times \hat{k} = \hat{j}\)