A force `vecF= (3hati+4hatj)` N acts on a body and displaces it by `vecS= (3hati+4hatj)m`. The work done (W= `vecF*vecS`) by the force is :

Options

A. 10J
B. 12 J
C. 19 J
D. 25 J

Explanation

To find the work done by the force \(\vec{F}\) on the body during the displacement \(\vec{S}\), we will use the formula for work done, which is given by the dot product of the force vector and the displacement vector:

\[ W = \vec{F} \cdot \vec{S} \]

### Step 1: Identify the vectors
Given:
\[ \vec{F} = 3\hat{i} + 4\hat{j} \quad \text{(in Newtons)} \]
\[ \vec{S} = 3\hat{i} + 4\hat{j} \quad \text{(in meters)} \]

### Step 2: Calculate the dot product
The dot product of two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j}\) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 \]

Applying this to our vectors:
\[ W = (3\hat{i} + 4\hat{j}) \cdot (3\hat{i} + 4\hat{j}) \]
\[ = 3 \cdot 3 + 4 \cdot 4 \]

### Step 3: Perform the multiplication
Calculating the individual products:
\[ = 9 + 16 \]

### Step 4: Sum the results
Now, add the results:
\[ W = 25 \, \text{Joules} \]

### Final Answer
The work done by the force is:
\[ W = 25 \, \text{Joules} \]

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