The projection of `vec(A)` on `vec(B)` is :-

Options

A. `vec(A).vec(B)`
B. `vec(A).hat(B)`
C. `vec(B).vec(A)`
D. `hat(A).hat(B)`

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** onto vector **B** is a vector that represents how much of **A** lies in the direction of **B**.

2. **Identify the Angle**: Let θ be the angle between vector **A** and vector **B**.

3. **Use the Projection Formula**: The formula for the projection of vector **A** onto vector **B** is given by:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} \]
where \(\vec{A} \cdot \vec{B}\) is the dot product of vectors **A** and **B**, and \(|\vec{B}|\) is the magnitude of vector **B**.

4. **Calculate the Dot Product**: The dot product can be expressed as:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \]

5. **Substitute into the Projection Formula**: Substitute the dot product into the projection formula:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{|\vec{A}| |\vec{B}| \cos \theta}{|\vec{B}|^2} \vec{B} \]

6. **Simplify the Expression**: This simplifies to:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{|\vec{A}| \cos \theta}{|\vec{B}|} \vec{B} \]

7. **Final Result**: The magnitude of the projection of vector **A** on vector **B** can be expressed as:
\[ |\text{proj}_{\vec{B}} \vec{A}| = |\vec{A}| \cos \theta \]

### Final Answer:
The projection of vector **A** on vector **B** is given by the vector:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} \]