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

Options

A. 100 J
B. 200 J
C. 300 J
D. 250 J

Explanation

To solve the problem step by step, we will calculate the displacement of the particle and then use the work done formula.

### Step 1: Identify the initial and final position vectors
The initial position vector \( \vec{r_1} \) is given as:
\[ \vec{r_1} = 3\hat{i} + 2\hat{j} - 6\hat{k} \]
The final position vector \( \vec{r_2} \) is given as:
\[ \vec{r_2} = 14\hat{i} + 13\hat{j} + 9\hat{k} \]

### Step 2: Calculate the displacement vector
The displacement vector \( \vec{s} \) can be calculated using the formula:
\[ \vec{s} = \vec{r_2} - \vec{r_1} \]
Substituting the values:
\[ \vec{s} = (14\hat{i} + 13\hat{j} + 9\hat{k}) - (3\hat{i} + 2\hat{j} - 6\hat{k}) \]
Calculating each component:
- For \( \hat{i} \): \( 14 - 3 = 11 \)
- For \( \hat{j} \): \( 13 - 2 = 11 \)
- For \( \hat{k} \): \( 9 - (-6) = 9 + 6 = 15 \)

Thus, the displacement vector is:
\[ \vec{s} = 11\hat{i} + 11\hat{j} + 15\hat{k} \]

### Step 3: Identify the force vector
The force vector \( \vec{F} \) is given as:
\[ \vec{F} = 4\hat{i} + 1\hat{j} + 3\hat{k} \]

### Step 4: Calculate the work done
The work done \( W \) is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{s} \]
Calculating the dot product:
\[ W = (4\hat{i} + 1\hat{j} + 3\hat{k}) \cdot (11\hat{i} + 11\hat{j} + 15\hat{k}) \]
Calculating each component:
- \( \hat{i} \) component: \( 4 \times 11 = 44 \)
- \( \hat{j} \) component: \( 1 \times 11 = 11 \)
- \( \hat{k} \) component: \( 3 \times 15 = 45 \)

Adding these components together:
\[ W = 44 + 11 + 45 = 100 \text{ Joules} \]

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