Two vectors `vecP and vecQ` are inclined to each other at angle `theta`. Which of the following is the unit vector perpendicular to `vecP and vecQ` ?

Options

A. `(vecPxx vecQ)/(P*Q)`
B. `(hatPxxhatQ)/(sintheta)`
C. `(hatPxxhatQ)/(PQ sin theta)`
D. `(hatPxx vecQ)/(PQ sin theta)`

Explanation

To find the unit vector that is perpendicular to two vectors \(\vec{P}\) and \(\vec{Q}\) inclined at an angle \(\theta\), we can follow these steps:

### Step 1: Understand the Cross Product
The cross product of two vectors \(\vec{P}\) and \(\vec{Q}\) gives a vector that is perpendicular to both \(\vec{P}\) and \(\vec{Q}\). Mathematically, this is expressed as:
\[ \vec{P} \times \vec{Q} \]
This resultant vector is perpendicular to the plane formed by \(\vec{P}\) and \(\vec{Q}\).

### Step 2: Calculate the Magnitude of the Cross Product
The magnitude of the cross product can be calculated using the formula:
\[ |\vec{P} \times \vec{Q}| = |\vec{P}| |\vec{Q}| \sin(\theta) \]
where \(|\vec{P}|\) and \(|\vec{Q}|\) are the magnitudes of vectors \(\vec{P}\) and \(\vec{Q}\), respectively, and \(\theta\) is the angle between them.

### Step 3: Define the Unit Vector
A unit vector in the direction of \(\vec{P} \times \vec{Q}\) is obtained by dividing the cross product by its magnitude:
\[ \hat{n} = \frac{\vec{P} \times \vec{Q}}{|\vec{P} \times \vec{Q}|} \]

### Step 4: Substitute the Magnitude
Substituting the magnitude from Step 2 into the unit vector formula gives:
\[ \hat{n} = \frac{\vec{P} \times \vec{Q}}{|\vec{P}| |\vec{Q}| \sin(\theta)} \]

### Step 5: Express the Unit Vector
Thus, the unit vector that is perpendicular to both \(\vec{P}\) and \(\vec{Q}\) can be expressed as:
\[ \hat{n} = \frac{\hat{P} \times \hat{Q}}{\sin(\theta)} \]
where \(\hat{P} = \frac{\vec{P}}{|\vec{P}|}\) and \(\hat{Q} = \frac{\vec{Q}}{|\vec{Q}|}\) are the unit vectors in the directions of \(\vec{P}\) and \(\vec{Q}\), respectively.

### Final Answer
The unit vector perpendicular to \(\vec{P}\) and \(\vec{Q}\) is:
\[ \hat{n} = \frac{\hat{P} \times \hat{Q}}{\sin(\theta)} \]