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 \(\overset{\rarr}{A} = 4\hat{i} + 3\hat{j}\) on vector \(\overset{\rarr}{B} = 3\hat{i} + 4\hat{j}\), we can follow these steps:

### Step 1: Calculate the dot product of vectors A and B
The dot product \(\overset{\rarr}{A} \cdot \overset{\rarr}{B}\) is calculated as follows:

\[ \overset{\rarr}{A} \cdot \overset{\rarr}{B} = (4\hat{i} + 3\hat{j}) \cdot (3\hat{i} + 4\hat{j}) = 4 \cdot 3 + 3 \cdot 4 = 12 + 12 = 24 \]

### Step 2: Calculate the magnitude of vector B
The magnitude of vector \(\overset{\rarr}{B}\) is given by:

\[ |\overset{\rarr}{B}| = \sqrt{(3^2 + 4^2)} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Use the formula for the projection of A on B
The projection of vector \(\overset{\rarr}{A}\) on vector \(\overset{\rarr}{B}\) is given by the formula:

\[ \text{Projection of } \overset{\rarr}{A} \text{ on } \overset{\rarr}{B} = \frac{\overset{\rarr}{A} \cdot \overset{\rarr}{B}}{|\overset{\rarr}{B}|} \]

Substituting the values we calculated:

\[ \text{Projection of } \overset{\rarr}{A} \text{ on } \overset{\rarr}{B} = \frac{24}{5} \]

### Final Answer
The projection of vector \(\overset{\rarr}{A}\) on vector \(\overset{\rarr}{B}\) is \(\frac{24}{5}\).

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