A paricle moves from a point `(-2 hat i + 5 hat j)` to `(4 hat j + 3 hat k)` when a foce of `(4 hat i + 3 hat j)N` is applied. How much work has been done by the force?

Options

A. `5 J`
B. `2J`
C. `8J`
D. `11J`

Explanation

To solve the problem of calculating the work done by the force on the particle, we will follow these steps:

### Step 1: Identify Initial and Final Positions
The initial position of the particle is given as:
\[ \mathbf{R_i} = -2\hat{i} + 5\hat{j} \]
The final position of the particle is given as:
\[ \mathbf{R_f} = 4\hat{j} + 3\hat{k} \]

### Step 2: Calculate the Displacement
Displacement \(\Delta \mathbf{r}\) is calculated using the formula:
\[ \Delta \mathbf{r} = \mathbf{R_f} - \mathbf{R_i} \]
Substituting the values:
\[ \Delta \mathbf{r} = (4\hat{j} + 3\hat{k}) - (-2\hat{i} + 5\hat{j}) \]
This simplifies to:
\[ \Delta \mathbf{r} = 4\hat{j} + 3\hat{k} + 2\hat{i} - 5\hat{j} \]
Combining like terms:
\[ \Delta \mathbf{r} = 2\hat{i} - 1\hat{j} + 3\hat{k} \]

### Step 3: Identify the Force
The force applied on the particle is given as:
\[ \mathbf{F} = 4\hat{i} + 3\hat{j} \]

### Step 4: Calculate the Work Done
The work done \(W\) by the force is given by the dot product of the force and the displacement:
\[ W = \mathbf{F} \cdot \Delta \mathbf{r} \]
Substituting the values:
\[ W = (4\hat{i} + 3\hat{j}) \cdot (2\hat{i} - 1\hat{j} + 3\hat{k}) \]
Calculating the dot product:
\[ W = 4 \cdot 2 + 3 \cdot (-1) + 0 \cdot 3 \]
This simplifies to:
\[ W = 8 - 3 + 0 = 5 \text{ Joules} \]

### Final Answer
The work done by the force is:
\[ \boxed{5 \text{ Joules}} \]

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