A particle moves from a point `(-2hat(i) + 5hat(j)) " to" " " (4hat(j) + 3hat(k))` when a force of `(4hat(i)+ 3hat(j))` N is force?

Options

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

Answer

To solve the problem step by step, we need to find the work done by the force on the particle as it moves from one point to another. Here’s how we can do it:

### Step 1: Identify the initial and final position vectors
The initial position vector \( \mathbf{r_1} \) is given as:
\[ \mathbf{r_1} = -2\hat{i} + 5\hat{j} \]
The final position vector \( \mathbf{r_2} \) is given as:
\[ \mathbf{r_2} = 4\hat{j} + 3\hat{k} \]

### Step 2: Calculate the displacement vector \( \mathbf{s} \)
The displacement vector \( \mathbf{s} \) can be calculated using the formula:
\[ \mathbf{s} = \mathbf{r_2} - \mathbf{r_1} \]
Substituting the values:
\[ \mathbf{s} = (4\hat{j} + 3\hat{k}) - (-2\hat{i} + 5\hat{j}) \]
\[ \mathbf{s} = 4\hat{j} + 3\hat{k} + 2\hat{i} - 5\hat{j} \]
\[ \mathbf{s} = 2\hat{i} - j + 3\hat{k} \]

### 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 \( W \)
The work done \( W \) by the force is given by the dot product of the force vector and the displacement vector:
\[ W = \mathbf{F} \cdot \mathbf{s} \]
Substituting the values:
\[ W = (4\hat{i} + 3\hat{j}) \cdot (2\hat{i} - 1\hat{j} + 3\hat{k}) \]
Calculating the dot product:
\[ W = (4 \cdot 2) + (3 \cdot -1) + (0 \cdot 3) \]
\[ W = 8 - 3 + 0 \]
\[ W = 5 \, \text{Joules} \]

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

---