A body is displaced from, `A=(2m, 4m, -6 m)` to `r_(B)=(6hati-4hatj + 2hatk)m` under a constant force, ` F=(2hati + 3hatj-hatk)N`. Find the work done.

Explanation

To solve the problem of finding the work done when a body is displaced under a constant force, we can follow these steps:

### Step 1: Identify the initial and final position vectors
The initial position vector \( \mathbf{A} \) and the final position vector \( \mathbf{B} \) are given as:
- \( \mathbf{A} = (2 \hat{i} + 4 \hat{j} - 6 \hat{k}) \, \text{m} \)
- \( \mathbf{B} = (6 \hat{i} - 4 \hat{j} + 2 \hat{k}) \, \text{m} \)

### Step 2: Calculate the displacement vector
The displacement vector \( \mathbf{S} \) can be calculated using the formula:
\[ \mathbf{S} = \mathbf{B} - \mathbf{A} \]
Calculating this gives:
\[ \mathbf{S} = (6 \hat{i} - 4 \hat{j} + 2 \hat{k}) - (2 \hat{i} + 4 \hat{j} - 6 \hat{k}) \]
\[ = (6 - 2) \hat{i} + (-4 - 4) \hat{j} + (2 + 6) \hat{k} \]
\[ = 4 \hat{i} - 8 \hat{j} + 8 \hat{k} \]

### Step 3: Identify the force vector
The force vector \( \mathbf{F} \) is given as:
\[ \mathbf{F} = (2 \hat{i} + 3 \hat{j} - 1 \hat{k}) \, \text{N} \]

### Step 4: Calculate the work done
The work done \( W \) is calculated using the dot product of the force vector and the displacement vector:
\[ W = \mathbf{F} \cdot \mathbf{S} \]
Calculating the dot product:
\[ W = (2 \hat{i} + 3 \hat{j} - 1 \hat{k}) \cdot (4 \hat{i} - 8 \hat{j} + 8 \hat{k}) \]
\[ = (2 \cdot 4) + (3 \cdot -8) + (-1 \cdot 8) \]
\[ = 8 - 24 - 8 \]
\[ = -24 \, \text{J} \]

### Conclusion
The work done is:
\[ \boxed{-24 \, \text{J}} \]

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