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

Options

A. 18J
B. 11J
C. 5J
D. 2J

Explanation

To solve the problem of how much work has been 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 positions
The initial position of the particle is given as:
\[ \mathbf{d_1} = -2\hat{i} + 5\hat{j} \]
The final position of the particle is:
\[ \mathbf{d_2} = 4\hat{i} + 3\hat{j} \]

### Step 2: Calculate the displacement vector
The displacement vector \(\mathbf{d}\) can be calculated using the formula:
\[ \mathbf{d} = \mathbf{d_2} - \mathbf{d_1} \]
Substituting the values:
\[ \mathbf{d} = (4\hat{i} + 3\hat{j}) - (-2\hat{i} + 5\hat{j}) \]
This simplifies to:
\[ \mathbf{d} = 4\hat{i} + 3\hat{j} + 2\hat{i} - 5\hat{j} = (4 + 2)\hat{i} + (3 - 5)\hat{j} = 6\hat{i} - 2\hat{j} \]

### Step 3: Identify the force vector
The force vector 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 calculated using the dot product of the force and displacement vectors:
\[ W = \mathbf{F} \cdot \mathbf{d} \]
Substituting the values:
\[ W = (4\hat{i} + 3\hat{j}) \cdot (6\hat{i} - 2\hat{j}) \]

### Step 5: Perform the dot product
Using the properties of the dot product:
\[ W = (4 \cdot 6) + (3 \cdot -2) = 24 - 6 = 18 \]

### Step 6: State the final answer
Thus, the work done by the force is:
\[ W = 18 \text{ Joules} \]

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