A particle moves from position `3hati+2hatj-6hatk` to `14hati+13hatj+9hatk` due to a uniform force of `(4hati+hatj+3hatk)N`. If the displacement in meters then work done will be

Options

A. 200J
B. 100J
C. 300J
D. 500J

Answer

To solve the problem step by step, we will calculate the displacement of the particle and then use it to find the work done by the force.

### Step 1: Identify the Initial and Final Positions
The initial position \( \mathbf{r_i} \) of the particle is given as:
\[ \mathbf{r_i} = 3\hat{i} + 2\hat{j} - 6\hat{k} \]
The final position \( \mathbf{r_f} \) of the particle is given as:
\[ \mathbf{r_f} = 14\hat{i} + 13\hat{j} + 9\hat{k} \]

### Step 2: Calculate the Displacement Vector
The displacement \( \mathbf{s} \) is calculated as:
\[ \mathbf{s} = \mathbf{r_f} - \mathbf{r_i} \]
Substituting the values:
\[ \mathbf{s} = (14\hat{i} + 13\hat{j} + 9\hat{k}) - (3\hat{i} + 2\hat{j} - 6\hat{k}) \]
Now, simplify the expression:
\[ \mathbf{s} = (14 - 3)\hat{i} + (13 - 2)\hat{j} + (9 + 6)\hat{k} \]
\[ \mathbf{s} = 11\hat{i} + 11\hat{j} + 15\hat{k} \]

### Step 3: Identify the Force Vector
The force \( \mathbf{F} \) acting on the particle is given as:
\[ \mathbf{F} = 4\hat{i} + 1\hat{j} + 3\hat{k} \]

### Step 4: Calculate the Work Done
The work done \( W \) by the force is given by the dot product of the force and the displacement:
\[ W = \mathbf{F} \cdot \mathbf{s} \]
Substituting the values:
\[ W = (4\hat{i} + 1\hat{j} + 3\hat{k}) \cdot (11\hat{i} + 11\hat{j} + 15\hat{k}) \]
Calculating the dot product:
\[ W = (4 \cdot 11) + (1 \cdot 11) + (3 \cdot 15) \]
\[ W = 44 + 11 + 45 \]
\[ W = 100 \text{ Joules} \]

### Final Answer
The work done is:
\[ \boxed{100 \text{ Joules}} \]