Projection of `overset(rarr)P` on `overset(rarr)Q` is :

Options

A. `overset (rarr)P.hat Q`
B. `overset (rarr)P.hat Q`
C. `overset (rarr)Pxxhat Q`
D. `overset(rarr)Pxxhat Q`

Explanation

To find the projection of vector **P** on vector **Q**, we can follow these steps:

### Step 1: Understand the Concept of Projection
The projection of one vector onto another is a way of expressing how much of the first vector lies in the direction of the second vector.

### Step 2: Use the Formula for Projection
The formula for the projection of vector **P** onto vector **Q** is given by:

\[ \text{Projection of } \vec{P} \text{ on } \vec{Q} = \frac{\vec{P} \cdot \vec{Q}}{|\vec{Q}|^2} \vec{Q} \]

Where:
- \(\vec{P} \cdot \vec{Q}\) is the dot product of vectors **P** and **Q**.
- \(|\vec{Q}|\) is the magnitude of vector **Q**.

### Step 3: Calculate the Dot Product
The dot product \(\vec{P} \cdot \vec{Q}\) can be calculated as:

\[ \vec{P} \cdot \vec{Q} = |\vec{P}| |\vec{Q}| \cos(\theta) \]

Where \(\theta\) is the angle between the two vectors.

### Step 4: Substitute into the Projection Formula
Substituting the dot product into the projection formula, we get:

\[ \text{Projection of } \vec{P} \text{ on } \vec{Q} = \frac{|\vec{P}| |\vec{Q}| \cos(\theta)}{|\vec{Q}|^2} \vec{Q} \]

### Step 5: Simplify the Expression
This simplifies to:

\[ \text{Projection of } \vec{P} \text{ on } \vec{Q} = \frac{|\vec{P}| \cos(\theta)}{|\vec{Q}|} \vec{Q} \]

### Step 6: Express in Terms of Unit Vector
If we denote the unit vector in the direction of **Q** as \(\hat{Q} = \frac{\vec{Q}}{|\vec{Q}|}\), we can rewrite the projection as:

\[ \text{Projection of } \vec{P} \text{ on } \vec{Q} = (\vec{P} \cdot \hat{Q}) \hat{Q} \]

### Conclusion
Thus, the projection of vector **P** on vector **Q** can be expressed as:

\[ \text{Projection of } \vec{P} \text{ on } \vec{Q} = \vec{P} \cdot \hat{Q} \hat{Q} \]