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 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:

**Step 1: Identify the Force and Displacement Vectors**

Given:
- Force vector \( \mathbf{F} = 3 \hat{i} + 2 \hat{j} \) N
- Displacement vector \( \mathbf{d} = 2 \hat{i} - 3 \hat{j} \) m

**Step 2: Write the Formula for Work Done**

The work done \( W \) is calculated using the dot product:
\[ W = \mathbf{F} \cdot \mathbf{d} \]

**Step 3: Calculate the Dot Product**

The dot product of two vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} \) is given by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2 \]

Applying this to our vectors:
\[ \mathbf{F} \cdot \mathbf{d} = (3)(2) + (2)(-3) \]

Calculating each term:
- First term: \( 3 \times 2 = 6 \)
- Second term: \( 2 \times -3 = -6 \)

Now, combine these results:
\[ W = 6 - 6 = 0 \]

**Step 4: Conclusion**

The work done is:
\[ W = 0 \text{ joules} \]

### Final Answer:
The work done is **0 joules**.

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