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

Options

A. 1 J
B. 2 J
C. 3 J
D. 2.5 J

Explanation

To solve the problem step by step, we will follow these steps:

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

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

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

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

### Step 5: Convert the work done to centimeters
Since the displacement was given in centimeters, we need to convert the work done from Joules to the appropriate unit. Since \( 1 \text{ Joule} = 10^4 \text{ cm} \cdot \text{N} \), the work done remains:
\[ W = 100 \text{ Joules} \]

### Final Answer
The work done is \( 100 \text{ Joules} \).

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