A force `vecF=(5hati+3hatj)` Newton is applied over a particle which displaces it from its origin to the point `vecr=(2hati-1hatj)` metres. The work done on the particle is

Options

A. `-7J`
B. `+13J`
C. `+7J`
D. `+11J`

Answer

To find the work done on the particle by the force, we can use the formula for work done, which is given by the dot product of the force vector and the displacement vector.

### Step-by-Step Solution:

1. **Identify the Force and Displacement Vectors:**
- The force vector is given as:
\[ \vec{F} = 5 \hat{i} + 3 \hat{j} \text{ Newton} \]
- The displacement vector is given as:
\[ \vec{r} = 2 \hat{i} - 1 \hat{j} \text{ metres} \]

2. **Calculate the Dot Product of the Force and Displacement Vectors:**
- The dot product \(\vec{F} \cdot \vec{r}\) is calculated as follows:
\[ \vec{F} \cdot \vec{r} = (5 \hat{i} + 3 \hat{j}) \cdot (2 \hat{i} - 1 \hat{j}) \]
- Using the properties of dot product:
\[ \vec{F} \cdot \vec{r} = (5 \cdot 2) + (3 \cdot -1) \]

3. **Perform the Multiplications:**
- Calculate \(5 \cdot 2\) and \(3 \cdot -1\):
\[ 5 \cdot 2 = 10 \]
\[ 3 \cdot -1 = -3 \]

4. **Add the Results of the Dot Product:**
- Now, add the results from the previous step:
\[ \vec{F} \cdot \vec{r} = 10 - 3 = 7 \]

5. **Conclusion:**
- The work done \(W\) on the particle is:
\[ W = 7 \text{ Joules} \]

### Final Answer:
The work done on the particle is **7 Joules**.