A particle moves from position `r_1=(3hati+2hatj-6hatk)` m to position `r_2=(14hati+13hatj+9hatk)` m under the action of a force `(4hati+hatj-3hatk)` N , then the work done is

Options

A. 46 J
B. 56 J
C. 86 J
D. 10 J

Answer

To find the work done by the force as a particle moves from position \( r_1 \) to position \( r_2 \), we can follow these steps:

### Step 1: Identify the position vectors
Given:
- \( r_1 = 3\hat{i} + 2\hat{j} - 6\hat{k} \) m
- \( r_2 = 14\hat{i} + 13\hat{j} + 9\hat{k} \) m

### Step 2: Calculate the displacement vector
The displacement vector \( \Delta r \) can be calculated as:
\[ \Delta r = r_2 - r_1 \]
Substituting the values:
\[ \Delta r = (14\hat{i} + 13\hat{j} + 9\hat{k}) - (3\hat{i} + 2\hat{j} - 6\hat{k}) \]
\[ = (14 - 3)\hat{i} + (13 - 2)\hat{j} + (9 + 6)\hat{k} \]
\[ = 11\hat{i} + 11\hat{j} + 15\hat{k} \]

### Step 3: Identify the force vector
The force vector \( \vec{F} \) is given as:
\[ \vec{F} = 4\hat{i} + 1\hat{j} - 3\hat{k} \text{ N} \]

### Step 4: Calculate the work done
The work done \( W \) by the force during the displacement is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \Delta r \]
Calculating the dot product:
\[ W = (4\hat{i} + 1\hat{j} - 3\hat{k}) \cdot (11\hat{i} + 11\hat{j} + 15\hat{k}) \]
\[ = (4 \cdot 11) + (1 \cdot 11) + (-3 \cdot 15) \]
\[ = 44 + 11 - 45 \]
\[ = 55 - 45 \]
\[ = 10 \text{ Joules} \]

### Conclusion
The work done by the force is \( 10 \) Joules.

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