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-by-Step Solution:

1. **Understand the Concept of Projection**:
The projection of one vector onto another is a measure of how much of the first vector lies in the direction of the second vector. In this case, we want to find the projection of vector **P** onto vector **Q**.

2. **Use the Projection Formula**:
The formula for the projection of vector **P** onto vector **Q** is given by:
\[ \text{Projection of } \overset{\rarr}{P} \text{ on } \overset{\rarr}{Q} = \frac{\overset{\rarr}{P} \cdot \overset{\rarr}{Q}}{|\overset{\rarr}{Q}|^2} \overset{\rarr}{Q} \]
where \( \overset{\rarr}{P} \cdot \overset{\rarr}{Q} \) is the dot product of vectors **P** and **Q**, and \( |\overset{\rarr}{Q}| \) is the magnitude of vector **Q**.

3. **Express the Projection in Terms of Unit Vector**:
The unit vector in the direction of **Q** is given by:
\[ \overset{\rarr}{Q}_{\text{cap}} = \frac{\overset{\rarr}{Q}}{|\overset{\rarr}{Q}|} \]
Therefore, we can rewrite the projection as:
\[ \text{Projection of } \overset{\rarr}{P} \text{ on } \overset{\rarr}{Q} = \overset{\rarr}{P} \cdot \overset{\rarr}{Q}_{\text{cap}} \]

4. **Final Expression**:
Thus, the projection of vector **P** on vector **Q** can be simplified to:
\[ \text{Projection of } \overset{\rarr}{P} \text{ on } \overset{\rarr}{Q} = \overset{\rarr}{P} \cdot \overset{\rarr}{Q}_{\text{cap}} \]

### Conclusion:
The projection of vector **P** on vector **Q** is given by the dot product of **P** with the unit vector of **Q**:
\[ \text{Projection of } \overset{\rarr}{P} \text{ on } \overset{\rarr}{Q} = \overset{\rarr}{P} \cdot \overset{\rarr}{Q}_{\text{cap}} \]