A force `vecF=6hati+2hatj+3hatk` acts on a particle and produces a displacement of `vecS=2hati-3hatj+xhatk`. If the work done is zero, the value of x is

Options

A. `-2`
B. 1/2
C. 6
D. 2

Explanation

To solve the problem, we need to find the value of \( x \) such that the work done by the force \( \vec{F} \) on the particle during the displacement \( \vec{S} \) is zero.

### Step-by-Step Solution:

1. **Identify the given vectors**:
- The force vector is given as:
\[ \vec{F} = 6 \hat{i} + 2 \hat{j} + 3 \hat{k} \]
- The displacement vector is given as:
\[ \vec{S} = 2 \hat{i} - 3 \hat{j} + x \hat{k} \]

2. **Understand the work done formula**:
- The work done \( W \) by a force \( \vec{F} \) during a displacement \( \vec{S} \) is given by the dot product:
\[ W = \vec{F} \cdot \vec{S} \]
- If the work done is zero, then:
\[ \vec{F} \cdot \vec{S} = 0 \]

3. **Calculate the dot product**:
- The dot product \( \vec{F} \cdot \vec{S} \) can be calculated as follows:
\[ \vec{F} \cdot \vec{S} = (6 \hat{i} + 2 \hat{j} + 3 \hat{k}) \cdot (2 \hat{i} - 3 \hat{j} + x \hat{k}) \]
- This expands to:
\[ = 6 \cdot 2 + 2 \cdot (-3) + 3 \cdot x \]
- Simplifying this gives:
\[ = 12 - 6 + 3x = 6 + 3x \]

4. **Set the dot product equal to zero**:
- Since the work done is zero:
\[ 6 + 3x = 0 \]

5. **Solve for \( x \)**:
- Rearranging the equation:
\[ 3x = -6 \]
- Dividing both sides by 3:
\[ x = -2 \]

### Final Answer:
The value of \( x \) is \( -2 \).