A particle moves from point P(1,2,3) to (2,1,4) under the action of a constant force `F=(2hati+hatj+hatk)N`. Work done by the force is

Options

A. 2 J
B. 4 J
C. 16 J
D. 8 J

Answer

To find the work done by the force when a particle moves from point P(1,2,3) to point Q(2,1,4) under the action of a constant force \( \mathbf{F} = 2\hat{i} + \hat{j} + \hat{k} \) N, we can follow these steps:

### Step 1: Determine the Displacement Vector
The displacement vector \( \mathbf{d} \) can be calculated by subtracting the initial position vector from the final position vector.

1. **Initial Position Vector \( \mathbf{P} \)**:
\[ \mathbf{P} = 1\hat{i} + 2\hat{j} + 3\hat{k} \]

2. **Final Position Vector \( \mathbf{Q} \)**:
\[ \mathbf{Q} = 2\hat{i} + 1\hat{j} + 4\hat{k} \]

3. **Displacement Vector \( \mathbf{d} \)**:
\[ \mathbf{d} = \mathbf{Q} - \mathbf{P} = (2\hat{i} + 1\hat{j} + 4\hat{k}) - (1\hat{i} + 2\hat{j} + 3\hat{k}) \]
\[ \mathbf{d} = (2 - 1)\hat{i} + (1 - 2)\hat{j} + (4 - 3)\hat{k} = 1\hat{i} - 1\hat{j} + 1\hat{k} \]

### Step 2: Calculate Work Done
The work done \( W \) by the force is given by the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{d} \).

1. **Force Vector \( \mathbf{F} \)**:
\[ \mathbf{F} = 2\hat{i} + 1\hat{j} + 1\hat{k} \]

2. **Dot Product \( \mathbf{F} \cdot \mathbf{d} \)**:
\[ W = \mathbf{F} \cdot \mathbf{d} = (2\hat{i} + 1\hat{j} + 1\hat{k}) \cdot (1\hat{i} - 1\hat{j} + 1\hat{k}) \]
\[ = 2 \cdot 1 + 1 \cdot (-1) + 1 \cdot 1 \]
\[ = 2 - 1 + 1 = 2 \text{ joules} \]

### Final Answer
The work done by the force is \( \boxed{2 \text{ joules}} \).

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