A force `vec(F) = (5hat(i) + 3hat(j)) N` is applied over a particle which displaces it from its origin to the point `vec(r ) = (2hat(i) - 1hat(j))` meter. The work done on the particle is :

Options

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

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 Vector and Displacement Vector:**
- The force vector is given as:
\[ \vec{F} = 5\hat{i} + 3\hat{j} \text{ N} \]
- The displacement vector from the origin to the point \((2\hat{i} - 1\hat{j})\) is:
\[ \vec{r} = 2\hat{i} - 1\hat{j} \text{ m} \]

2. **Calculate the Work Done:**
- The work done \(W\) is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{r} \]
- Substitute the vectors into the dot product:
\[ W = (5\hat{i} + 3\hat{j}) \cdot (2\hat{i} - 1\hat{j}) \]

3. **Perform the Dot Product Calculation:**
- The dot product is calculated as follows:
\[ W = (5 \cdot 2) + (3 \cdot -1) \]
- Calculate each term:
\[ W = 10 - 3 \]
- Therefore:
\[ W = 7 \text{ Joules} \]

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