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Question
A particle moves from a point \[\overrightarrow{r}_1 = \left( 2 m \right) \overrightarrow{ i } + \left( 3 m \right) \overrightarrow{ j } \] to another point
\[\overrightarrow{r}_2 = \left( 3 m \right) \overrightarrow{ i } + \left( 2 m \right) \overrightarrow{ j } \] acts on it. Find the work done by the force on the particle during the displacement.
Solution
Initial position vector, \[\vec{r_1} = 2 \vec{i} + 3 \vec{j}\] Final position vector, \[\vec{r}_2 = 3 \vec{i} + 2 \vec{j}\]
So, displacement vector,
\[\vec{r} = \vec{r}_2 - \vec{r}_1 \]
\[ = \left( 3 \vec{i} + 2 \vec{j} \right) - \left( 2 \vec{i} + \vec{j} \right)\]
\[ = \vec{i} - \vec{j} \]
\[\text{ Force acting on the particle } , \vec{F} = 5 \vec{i} + 5 \vec{j} \]
\[\text{ So, work done } = \vec{F} \cdot \vec{S} = 5 \times 1 + 5 \left( - 1 \right) = 0\]
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