A force `(vecF)=3hati+chatj+2hatk` acting on a particle causes a displacement: `(vecS)=-4hati+2hatj+3hatk` in its own direction. If the work done is 12 J, then the value of .c. is

Options

A. 0
B. 1
C. 9
D. 12

Explanation

To solve the problem, we need to find the value of \( c \) given the force vector \( \vec{F} = 3\hat{i} + c\hat{j} + 2\hat{k} \), the displacement vector \( \vec{S} = -4\hat{i} + 2\hat{j} + 3\hat{k} \), and the work done \( W = 12 \, \text{J} \).

### Step-by-Step Solution:

1. **Understand the Work Done Formula**:
The work done \( W \) by a force \( \vec{F} \) when it causes a displacement \( \vec{S} \) is given by the dot product:
\[ W = \vec{F} \cdot \vec{S} \]

2. **Write the Dot Product**:
Substitute the given vectors into the dot product formula:
\[ \vec{F} \cdot \vec{S} = (3\hat{i} + c\hat{j} + 2\hat{k}) \cdot (-4\hat{i} + 2\hat{j} + 3\hat{k}) \]

3. **Calculate the Dot Product**:
Using the properties of the dot product:
- \( \hat{i} \cdot \hat{i} = 1 \)
- \( \hat{j} \cdot \hat{j} = 1 \)
- \( \hat{k} \cdot \hat{k} = 1 \)
- \( \hat{i} \cdot \hat{j} = 0 \), \( \hat{j} \cdot \hat{k} = 0 \), \( \hat{k} \cdot \hat{i} = 0 \)

We can expand the dot product:
\[ \vec{F} \cdot \vec{S} = 3(-4) + c(2) + 2(3) \]
\[ = -12 + 2c + 6 \]
\[ = 2c - 6 \]

4. **Set the Dot Product Equal to Work Done**:
Since we know the work done is 12 J, we set up the equation:
\[ 2c - 6 = 12 \]

5. **Solve for \( c \)**:
Rearranging the equation:
\[ 2c = 12 + 6 \]
\[ 2c = 18 \]
\[ c = \frac{18}{2} = 9 \]

### Final Answer:
The value of \( c \) is \( 9 \).

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