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 when a force is applied, 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:
\[ \vec{F} = 5 \hat{i} + 3 \hat{j} \, \text{N} \]
- The displacement vector is given as:
\[ \vec{r} = 2 \hat{i} - 1 \hat{j} \, \text{m} \]

2. **Calculate the Dot Product:**
- The work done \( W \) is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{r} \]
- The dot product formula for two vectors \( \vec{A} = A_x \hat{i} + A_y \hat{j} \) and \( \vec{B} = B_x \hat{i} + B_y \hat{j} \) is:
\[ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y \]

3. **Substituting the Values:**
- For our vectors:
- \( A_x = 5 \), \( A_y = 3 \)
- \( B_x = 2 \), \( B_y = -1 \)
- Now substituting these values into the dot product formula:
\[ W = (5)(2) + (3)(-1) \]

4. **Perform the Calculations:**
- Calculate the first term:
\[ 5 \times 2 = 10 \]
- Calculate the second term:
\[ 3 \times -1 = -3 \]
- Now combine these results:
\[ W = 10 - 3 = 7 \, \text{J} \]

5. **Final Result:**
- The work done on the particle is:
\[ W = 7 \, \text{J} \]