What is the vector perpendicular to both the vector `hat(i) - hat(j) and hat(i)` ?

Options

A. `hat(i)`
B. `-hat(j)`
C. `hat(j)`
D. `hat(k)`

Answer

To find the vector that is perpendicular to both vectors \(\hat{i} - \hat{j}\) and \(\hat{i}\), we can use the concept of the cross product. The cross product of two vectors results in a vector that is perpendicular to both of the original vectors.

### Step-by-Step Solution:

1. **Identify the Vectors**:
- Let vector **A** = \(\hat{i} - \hat{j}\)
- Let vector **B** = \(\hat{i}\)

2. **Write the Vectors in Component Form**:
- Vector **A** = \(\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}\)
- Vector **B** = \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\)

3. **Calculate the Cross Product**:
- The cross product \(\mathbf{A} \times \mathbf{B}\) can be calculated using the determinant of a matrix:
\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 0 \\ 1 & 0 & 0 \end{vmatrix} \]

4. **Evaluate the Determinant**:
- Expanding the determinant:
\[ \mathbf{A} \times \mathbf{B} = \hat{i} \begin{vmatrix} -1 & 0 \\ 0 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 0 \\ 1 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -1 \\ 1 & 0 \end{vmatrix} \]
- Calculating the 2x2 determinants:
\[ = \hat{i}(0) - \hat{j}(0) + \hat{k}(-1 - (-1)) = \hat{k}(1) \]
- Thus, \(\mathbf{A} \times \mathbf{B} = \hat{k}\).

5. **Conclusion**:
- The vector that is perpendicular to both \(\hat{i} - \hat{j}\) and \(\hat{i}\) is \(\hat{k}\).

### Final Answer:
The vector perpendicular to both \(\hat{i} - \hat{j}\) and \(\hat{i}\) is \(\hat{k}\).