Which of the following is the unit vector perpendicular to ? `overset(rarr)A` and `overset(rarr)B` ?

Options

A. `(overset(rarr)A xx overset(rarr)B)/(AB sin theta)`
B. `(overset(rarr)A xx overset(rarr)B)/(AB cos theta)`
C. `(overset(rarr)A xx overset(rarr)B)/(|overset(rarr)A.overset(rarr)B|)`
D. `(overset(rarr)A xx overset(rarr)B)/(|overset(rarr)A||overset(rarr)B|)`

Explanation

To find the unit vector that is perpendicular to both vectors \( \vec{A} \) and \( \vec{B} \), we will follow these steps:

### Step 1: Understand the Cross Product
The vector that is perpendicular to both \( \vec{A} \) and \( \vec{B} \) can be found using the cross product. The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ \vec{C} = \vec{A} \times \vec{B} \]
This vector \( \vec{C} \) is perpendicular to both \( \vec{A} \) and \( \vec{B} \).

### Step 2: Calculate the Magnitude of the Cross Product
The magnitude of the cross product \( \vec{C} \) is given by:
\[ |\vec{C}| = |\vec{A}| |\vec{B}| \sin \theta \]
where \( \theta \) is the angle between vectors \( \vec{A} \) and \( \vec{B} \).

### Step 3: Find the Unit Vector
The unit vector \( \hat{C} \) in the direction of \( \vec{C} \) is given by:
\[ \hat{C} = \frac{\vec{C}}{|\vec{C}|} \]
Substituting the expression for \( \vec{C} \):
\[ \hat{C} = \frac{\vec{A} \times \vec{B}}{|\vec{A}| |\vec{B}| \sin \theta} \]

### Conclusion
Thus, the unit vector that is perpendicular to both \( \vec{A} \) and \( \vec{B} \) is:
\[ \hat{C} = \frac{\vec{A} \times \vec{B}}{|\vec{A}| |\vec{B}| \sin \theta} \]