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

### Step 1: Identify the initial and final positions
The particle moves from the point:
- Initial Position \( \mathbf{A} = -2\hat{i} + 5\hat{j} \)
- Final Position \( \mathbf{B} = 0\hat{i} + 4\hat{j} + 3\hat{k} \)

### Step 2: Calculate the displacement vector
The displacement vector \( \mathbf{d} \) can be calculated by subtracting the initial position from the final position:
\[ \mathbf{d} = \mathbf{B} - \mathbf{A} \]
Substituting the values:
\[ \mathbf{d} = (0\hat{i} + 4\hat{j} + 3\hat{k}) - (-2\hat{i} + 5\hat{j}) \]
This simplifies to:
\[ \mathbf{d} = (0 + 2)\hat{i} + (4 - 5)\hat{j} + (3 - 0)\hat{k} = 2\hat{i} - 1\hat{j} + 3\hat{k} \]

### Step 3: Write down the force vector
The force vector \( \mathbf{F} \) 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 vector and the displacement vector:
\[ W = \mathbf{F} \cdot \mathbf{d} \]
Substituting the vectors:
\[ W = (4\hat{i} + 3\hat{j}) \cdot (2\hat{i} - 1\hat{j} + 3\hat{k}) \]
Calculating the dot product:
\[ W = (4 \cdot 2) + (3 \cdot -1) + (0 \cdot 3) \]
\[ W = 8 - 3 + 0 = 5 \text{ joules} \]

### Final Answer
The work done by the force is \( 5 \text{ joules} \).

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