The work done by the force `4hati-3hatj +2hatk` in moving a particle along a straight line from the point (3, 2, - 1) to (2-1,4) is

Options

A. 0 units
B. 4 units
C. 15 units
D. 19 units

Answer

To find the work done by the force \( \vec{F} = 4\hat{i} - 3\hat{j} + 2\hat{k} \) in moving a particle from point \( P(3, 2, -1) \) to point \( Q(2, -1, 4) \), we can follow these steps:

### Step 1: Determine the position vectors
The position vector of point \( P \) is:
\[ \vec{r}_P = 3\hat{i} + 2\hat{j} - 1\hat{k} \]
The position vector of point \( Q \) is:
\[ \vec{r}_Q = 2\hat{i} - 1\hat{j} + 4\hat{k} \]

### Step 2: Calculate the displacement vector
The displacement vector \( \vec{d} \) from \( P \) to \( Q \) is given by:
\[ \vec{d} = \vec{r}_Q - \vec{r}_P \]
Calculating this gives:
\[ \vec{d} = (2\hat{i} - 1\hat{j} + 4\hat{k}) - (3\hat{i} + 2\hat{j} - 1\hat{k}) \]
\[ = (2 - 3)\hat{i} + (-1 - 2)\hat{j} + (4 + 1)\hat{k} \]
\[ = -1\hat{i} - 3\hat{j} + 5\hat{k} \]

### Step 3: Calculate the work done
The work done \( W \) by the force along the displacement is given by the dot product:
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the values of \( \vec{F} \) and \( \vec{d} \):
\[ W = (4\hat{i} - 3\hat{j} + 2\hat{k}) \cdot (-1\hat{i} - 3\hat{j} + 5\hat{k}) \]
Calculating the dot product:
\[ W = 4(-1) + (-3)(-3) + 2(5) \]
\[ = -4 + 9 + 10 \]
\[ = 15 \]

### Conclusion
The work done by the force in moving the particle from point \( P \) to point \( Q \) is:
\[ W = 15 \text{ Joules} \]