An object is moving along a straight line path from P to Q under the action of a force `vec(F)=(4 hati-3hatj+hat2k)N`. If the co-ordinate of P and Q in meters are (3,2,-1) and (2,-1,4) respectively. Then the work done by the force is

Options

A. `-15 J`
B. `+15 J`
C. `1015 J`
D. `(4hati-3hatj+2hatk)`

Explanation

To solve the problem of calculating the work done by the force as the object moves from point P to point Q, we will follow these steps:

### Step 1: Identify the coordinates of points P and Q
The coordinates of point P are given as (3, 2, -1) and the coordinates of point Q are (2, -1, 4).

### Step 2: Write the force vector
The force vector is given as:
\[ \vec{F} = 4 \hat{i} - 3 \hat{j} + 2 \hat{k} \text{ N} \]

### Step 3: Calculate the displacement vector
The displacement vector \(\vec{s}\) can be calculated by subtracting the coordinates of point P from point Q:
\[ \vec{s} = \vec{Q} - \vec{P} = (2 \hat{i} - 1 \hat{j} + 4 \hat{k}) - (3 \hat{i} + 2 \hat{j} - 1 \hat{k}) \]
Calculating this gives:
\[ \vec{s} = (2 - 3) \hat{i} + (-1 - 2) \hat{j} + (4 + 1) \hat{k} = -1 \hat{i} - 3 \hat{j} + 5 \hat{k} \]

### Step 4: Calculate the work done
The work done \(W\) by the force is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{s} = (4 \hat{i} - 3 \hat{j} + 2 \hat{k}) \cdot (-1 \hat{i} - 3 \hat{j} + 5 \hat{k}) \]
Calculating the dot product:
\[ W = (4)(-1) + (-3)(-3) + (2)(5) = -4 + 9 + 10 \]
Thus,
\[ W = 15 \text{ Joules} \]

### Final Answer
The work done by the force is \(15 \text{ Joules}\).

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