A particle moves from a point `(-2hati+5hatj)` to `(4hatj+3hatk)` when a force of `(4hati+3hatj)N` is applied. How much work has been done by the force ?

Options

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

Answer

To find the work done by the force when a particle moves from one point to another, we can follow these steps:

### Step 1: Identify the initial and final position vectors.
- The initial position vector \( \mathbf{r_i} \) is given as \( -2\hat{i} + 5\hat{j} \).
- The final position vector \( \mathbf{r_f} \) is given as \( 4\hat{j} + 3\hat{k} \).

### Step 2: Calculate the displacement vector \( \mathbf{s} \).
The displacement vector \( \mathbf{s} \) can be calculated using the formula:
\[ \mathbf{s} = \mathbf{r_f} - \mathbf{r_i} \]
Substituting the values:
\[ \mathbf{s} = (4\hat{j} + 3\hat{k}) - (-2\hat{i} + 5\hat{j}) \]
\[ = 0\hat{i} + (4 - 5)\hat{j} + 3\hat{k} \]
\[ = 2\hat{i} - 1\hat{j} + 3\hat{k} \]

### Step 3: Identify the force vector \( \mathbf{F} \).
The force vector \( \mathbf{F} \) is given as:
\[ \mathbf{F} = 4\hat{i} + 3\hat{j} \]

### Step 4: Calculate the work done \( W \) using the dot product.
The work done by the force is given by the formula:
\[ W = \mathbf{F} \cdot \mathbf{s} \]
Calculating the dot product:
\[ W = (4\hat{i} + 3\hat{j}) \cdot (2\hat{i} - 1\hat{j} + 3\hat{k}) \]
Using the properties of the dot product:
\[ W = (4 \cdot 2) + (3 \cdot -1) + (0 \cdot 3) \]
\[ = 8 - 3 + 0 \]
\[ = 5 \text{ Joules} \]

### Conclusion
The work done by the force is \( 5 \) Joules.

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