A particle is displaced from a position `(2 vec i - vec j + vec k)` metre to another position `(3 vec i + 2 vec j -2 vec k)` metre under the action of force `(2 vec i + vec j -vec k)`N. Work done by the force is

Options

A. `8 J`
B. `10 J`
C. `12 J`
D. `36 J`

Answer

To find the work done by the force on the particle, we will follow these steps:

### Step 1: Identify the initial and final position vectors
The initial position vector \( \vec{r_i} \) is given as:
\[ \vec{r_i} = 2 \vec{i} - \vec{j} + \vec{k} \]
The final position vector \( \vec{r_f} \) is given as:
\[ \vec{r_f} = 3 \vec{i} + 2 \vec{j} - 2 \vec{k} \]

### Step 2: Calculate the displacement vector \( \vec{d} \)
The displacement vector \( \vec{d} \) can be calculated by subtracting the initial position vector from the final position vector:
\[ \vec{d} = \vec{r_f} - \vec{r_i} \]
Substituting the values:
\[ \vec{d} = (3 \vec{i} + 2 \vec{j} - 2 \vec{k}) - (2 \vec{i} - \vec{j} + \vec{k}) \]
Now, perform the subtraction:
\[ \vec{d} = (3 - 2) \vec{i} + (2 + 1) \vec{j} + (-2 - 1) \vec{k} \]
This simplifies to:
\[ \vec{d} = 1 \vec{i} + 3 \vec{j} - 3 \vec{k} \]

### Step 3: Identify the force vector \( \vec{F} \)
The force vector \( \vec{F} \) is given as:
\[ \vec{F} = 2 \vec{i} + \vec{j} - \vec{k} \]

### Step 4: Calculate the work done \( W \)
The work done by the force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the values:
\[ W = (2 \vec{i} + \vec{j} - \vec{k}) \cdot (1 \vec{i} + 3 \vec{j} - 3 \vec{k}) \]
Calculating the dot product:
\[ W = (2 \cdot 1) + (1 \cdot 3) + (-1 \cdot -3) \]
This simplifies to:
\[ W = 2 + 3 + 3 = 8 \text{ joules} \]

### Final Answer
The work done by the force is:
\[ \boxed{8 \text{ joules}} \]