A force `barf=(hati-3hatj+2hatk)N` moves a particle from `bar r_(1)=2hati+7hatj+4hatk)m` to `(barr_(2)=(5hati+2hatj+8hatk)m`. The work done by the force is

Options

A. 18J
B. 28J
C. 38J
D. 48J

Answer

To solve the problem, we need to calculate the work done by the force when it moves a particle from position \( \mathbf{r_1} \) to \( \mathbf{r_2} \). The work done by a force is given by the dot product of the force vector and the displacement vector.

### Step-by-Step Solution:

1. **Identify the Force Vector and Position Vectors:**
- The force vector is given as:
\[ \mathbf{F} = \hat{i} - 3\hat{j} + 2\hat{k} \, \text{N} \]
- The initial position vector is:
\[ \mathbf{r_1} = 2\hat{i} + 7\hat{j} + 4\hat{k} \, \text{m} \]
- The final position vector is:
\[ \mathbf{r_2} = 5\hat{i} + 2\hat{j} + 8\hat{k} \, \text{m} \]

2. **Calculate the Displacement Vector:**
- The displacement vector \( \Delta \mathbf{r} \) is calculated as:
\[ \Delta \mathbf{r} = \mathbf{r_2} - \mathbf{r_1} \]
- Performing the subtraction:
\[ \Delta \mathbf{r} = (5\hat{i} + 2\hat{j} + 8\hat{k}) - (2\hat{i} + 7\hat{j} + 4\hat{k}) \]
\[ = (5 - 2)\hat{i} + (2 - 7)\hat{j} + (8 - 4)\hat{k} \]
\[ = 3\hat{i} - 5\hat{j} + 4\hat{k} \, \text{m} \]

3. **Calculate the Work Done:**
- The work done \( W \) is given by the dot product of the force vector and the displacement vector:
\[ W = \mathbf{F} \cdot \Delta \mathbf{r} \]
- Substituting the values:
\[ W = (\hat{i} - 3\hat{j} + 2\hat{k}) \cdot (3\hat{i} - 5\hat{j} + 4\hat{k}) \]
- Calculating the dot product:
\[ W = (1 \cdot 3) + (-3 \cdot -5) + (2 \cdot 4) \]
\[ = 3 + 15 + 8 \]
\[ = 26 \, \text{J} \]

### Final Answer:
The work done by the force is \( \boxed{26 \, \text{J}} \).