Write the projection of vector `veca" on "vecb`.

Answer

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

### Step 1: Understand the Concept of Projection
The projection of vector **A** onto vector **B** is essentially the component of vector **A** that points in the direction of vector **B**.

### Step 2: Use the Projection Formula
The formula for the projection of vector **A** on vector **B** is given by:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\vec{B} \cdot \vec{B}} \vec{B} \]
Where:
- \(\vec{A} \cdot \vec{B}\) is the dot product of vectors **A** and **B**.
- \(\vec{B} \cdot \vec{B}\) is the dot product of vector **B** with itself, which gives the magnitude squared of vector **B**.

### Step 3: Calculate the Dot Product
Calculate \(\vec{A} \cdot \vec{B}\):
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \]
Where \(\theta\) is the angle between vectors **A** and **B**.

### Step 4: Substitute into the Projection Formula
Substituting the dot product into the projection formula:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{|\vec{A}| |\vec{B}| \cos(\theta)}{|\vec{B}|^2} \vec{B} \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{|\vec{A}| \cos(\theta)}{|\vec{B}|} \vec{B} \]
This expression gives us the projection of vector **A** onto vector **B**.

### Final Expression
Thus, the projection of vector **A** on vector **B** is:
\[ \text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2} \vec{B} \]

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