A particle is displaced from a position `2hati-hatj+hatk (m)` to another position `3hati+2hatj-2hatk (m)` under the action of a force `2hati+hatj-hatk(N)`. The 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 during its displacement, we will follow these steps:

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

### Step 2: Calculate the displacement vector
The displacement vector \( \vec{S} \) can be calculated as:
\[ \vec{S} = \vec{r_f} - \vec{r_i} \]
Substituting the values:
\[ \vec{S} = (3\hat{i} + 2\hat{j} - 2\hat{k}) - (2\hat{i} - \hat{j} + \hat{k}) \]
\[ \vec{S} = (3\hat{i} - 2\hat{i}) + (2\hat{j} + \hat{j}) + (-2\hat{k} - \hat{k}) \]
\[ \vec{S} = \hat{i} + 3\hat{j} - 3\hat{k} \, \text{(m)} \]

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

### Step 4: Calculate the work done by the force
The work done \( W \) by a constant force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{S} \]
Calculating the dot product:
\[ W = (2\hat{i} + \hat{j} - \hat{k}) \cdot (\hat{i} + 3\hat{j} - 3\hat{k}) \]
\[ W = (2 \cdot 1) + (1 \cdot 3) + (-1 \cdot -3) \]
\[ W = 2 + 3 + 3 \]
\[ W = 8 \, \text{Joules} \]

### Final Answer
The work done by the force is \( 8 \, \text{J} \).

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