What is the projection of `vecA` on `vecB` ?

Options

A. `vecA*vecB`
B. `vecA*hatB`
C. `vecB*vecA`
D. `hatA*hatB`

Explanation

To find the projection of vector **A** on vector **B**, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Projection**:
The projection of vector **A** on vector **B** is a vector that represents how much of **A** lies in the direction of **B**.

2. **Use the Formula for Projection**:
The projection of vector **A** on vector **B** can be mathematically expressed as:
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} \]
where \(\vec{A} \cdot \vec{B}\) is the dot product of vectors **A** and **B**, and \(|\vec{B}|\) is the magnitude of vector **B**.

3. **Calculate the Dot Product**:
The dot product \(\vec{A} \cdot \vec{B}\) can also be expressed in terms of the magnitudes of the vectors and the cosine of the angle \(\theta\) between them:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]

4. **Substitute the Dot Product into the Projection Formula**:
Substitute the expression for the dot product into the projection formula:
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{|\vec{A}| |\vec{B}| \cos \theta}{|\vec{B}|^2} \vec{B} \]

5. **Simplify the Expression**:
The \(|\vec{B}|\) in the numerator and denominator cancels out:
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{|\vec{A}| \cos \theta}{|\vec{B}|} \vec{B} \]

6. **Final Result**:
Thus, the projection of vector **A** on vector **B** can be expressed as:
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = |\vec{A}| \cos \theta \hat{B} \]
where \(\hat{B}\) is the unit vector in the direction of vector **B**.