A force `(3hat(i) + 2hat(j)) N` displaces an object through a distance `(2hat(i) - 3hat(j))`m. The work done is :

Options

A. zero
B. 12 J
C. 5 J
D. 13 J

Answer

To find the work done by a force when an object is displaced, we 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 \( \vec{F} = 3\hat{i} + 2\hat{j} \) N.
- The displacement vector \( \vec{d} = 2\hat{i} - 3\hat{j} \) m.

2. **Use the Dot Product Formula**:
The work done \( W \) is calculated using the dot product:
\[ W = \vec{F} \cdot \vec{d} \]

3. **Calculate the Dot Product**:
The dot product of two vectors \( \vec{A} = a_1\hat{i} + b_1\hat{j} \) and \( \vec{B} = a_2\hat{i} + b_2\hat{j} \) is given by:
\[ \vec{A} \cdot \vec{B} = a_1a_2 + b_1b_2 \]
Applying this to our vectors:
\[ \vec{F} \cdot \vec{d} = (3)(2) + (2)(-3) \]

4. **Perform the Multiplication**:
- Calculate \( 3 \times 2 = 6 \)
- Calculate \( 2 \times -3 = -6 \)

5. **Add the Results**:
\[ W = 6 + (-6) = 6 - 6 = 0 \]

6. **Conclusion**:
The work done \( W \) is \( 0 \) Joules.

### Final Answer:
The work done is \( 0 \) Joules.