1. **State the problem:** Calculate the work done by the force $ \vec{F} = 3\hat{i} + 3\hat{j} $ when the displacement is $ \vec{x} = 4\hat{i} - \hat{j} $. 2. **Formula used:** Work done by a force is given by the dot product of force and displacement vectors: $$ W = \vec{F} \cdot \vec{x} $$ 3. **Dot product rule:** For vectors $ \vec{A} = a_1\hat{i} + a_2\hat{j} $ and $ \vec{B} = b_1\hat{i} + b_2\hat{j} $, $$ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 $$ 4. **Calculate the dot product:** $$ W = (3)(4) + (3)(-1) $$ $$ W = 12 - 3 $$ $$ W = 9 $$ 5. **Interpretation:** The work done by the force is 9 joules. This means the force has done 9 units of work in moving the object along the displacement vector.