What is the projection of vector `overset(rarr)A=4hat I +3 hatj` on vector `overset(rarr)B=3hat I +4 hat j` ?

Options

A. 10
B. `24//5`
C. zero
D. `12hat i+12hatj`

Explanation

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

### Step 1: Identify the vectors
We have:
- Vector **A** = \( 4\hat{i} + 3\hat{j} \)
- Vector **B** = \( 3\hat{i} + 4\hat{j} \)

### Step 2: Use the projection formula
The projection of vector **A** onto vector **B** is given by the formula:
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \]
where \( \vec{A} \cdot \vec{B} \) is the dot product of the vectors and \( |\vec{B}| \) is the magnitude of vector **B**.

### Step 3: Calculate the dot product \( \vec{A} \cdot \vec{B} \)
The dot product is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (4\hat{i} + 3\hat{j}) \cdot (3\hat{i} + 4\hat{j}) = 4 \cdot 3 + 3 \cdot 4 = 12 + 12 = 24 \]

### Step 4: Calculate the magnitude of vector **B**
The magnitude of vector **B** is calculated using the formula:
\[ |\vec{B}| = \sqrt{(3^2 + 4^2)} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 5: Calculate the projection
Now we can substitute the values into the projection formula:
\[ \text{Projection of } \vec{A} \text{ on } \vec{B} = \frac{24}{5} \]

### Final Answer
Thus, the projection of vector **A** on vector **B** is:
\[ \frac{24}{5} \]