A force `vec(F)=3hat(i)+chat(j)+2hat(k)` acting on a particle causes a displacement `vec(d)=-4hat(i)+2hat(j)+3hat(k)`. If the work done is 6J then the value of 'c' is

Options

A. 12
B. 0
C. 6
D. 3

Explanation

To find the value of 'c' in the given problem, we will use the formula for work done, which is given by the dot product of the force vector and the displacement vector.

### Step-by-Step Solution:

1. **Identify the Force and Displacement Vectors:**
- The force vector is given as \(\vec{F} = 3\hat{i} + c\hat{j} + 2\hat{k}\).
- The displacement vector is given as \(\vec{d} = -4\hat{i} + 2\hat{j} + 3\hat{k}\).

2. **Use the Work Done Formula:**
- The work done \(W\) is calculated using the dot product:
\[ W = \vec{F} \cdot \vec{d} \]

3. **Calculate the Dot Product:**
- The dot product \(\vec{F} \cdot \vec{d}\) is calculated as follows:
\[ \vec{F} \cdot \vec{d} = (3\hat{i} + c\hat{j} + 2\hat{k}) \cdot (-4\hat{i} + 2\hat{j} + 3\hat{k}) \]
- This expands to:
\[ = 3 \cdot (-4) + c \cdot 2 + 2 \cdot 3 \]
- Simplifying this gives:
\[ = -12 + 2c + 6 \]
- Therefore, the expression for work done becomes:
\[ W = -12 + 2c + 6 = 2c - 6 \]

4. **Set the Work Done Equal to 6 Joules:**
- We know from the problem that the work done is 6J:
\[ 2c - 6 = 6 \]

5. **Solve for 'c':**
- Rearranging the equation gives:
\[ 2c = 6 + 6 \]
\[ 2c = 12 \]
\[ c = \frac{12}{2} = 6 \]

6. **Final Answer:**
- The value of \(c\) is \(6\).