A particle is displaced from position `vec(r )_(1) = (3hat(i) + 2hat(j) + 6hat(k))` to another position `vec(r )_(2)= (14hat(i) + 13hat(j) + 9hat(k))` under the impact of a force `5 hat(i)N`. The work done will be given by

Options

A. 55J
B. 65J
C. 45J
D. 75J

Explanation

To solve the problem of calculating the work done when a particle is displaced under the influence of a force, we can follow these steps:

### Step 1: Identify the initial and final positions
The initial position of the particle is given as:
\[ \vec{r}_1 = 3\hat{i} + 2\hat{j} + 6\hat{k} \]
The final position of the particle is:
\[ \vec{r}_2 = 14\hat{i} + 13\hat{j} + 9\hat{k} \]

### Step 2: Calculate the displacement vector
The displacement vector \(\vec{r}\) can be calculated by subtracting the initial position from the final position:
\[ \vec{r} = \vec{r}_2 - \vec{r}_1 = (14\hat{i} + 13\hat{j} + 9\hat{k}) - (3\hat{i} + 2\hat{j} + 6\hat{k}) \]
Calculating this gives:
\[ \vec{r} = (14 - 3)\hat{i} + (13 - 2)\hat{j} + (9 - 6)\hat{k} = 11\hat{i} + 11\hat{j} + 3\hat{k} \]

### Step 3: Identify the force vector
The force acting on the particle is given as:
\[ \vec{F} = 5\hat{i} \text{ N} \]

### Step 4: Calculate the work done
The work done \(W\) by the force when the particle is displaced can be calculated using the dot product of the force and the displacement vectors:
\[ W = \vec{F} \cdot \vec{r} \]
Substituting the values:
\[ W = (5\hat{i}) \cdot (11\hat{i} + 11\hat{j} + 3\hat{k}) \]
Using the properties of the dot product:
\[ W = 5 \cdot 11 (\hat{i} \cdot \hat{i}) + 5 \cdot 11 (\hat{i} \cdot \hat{j}) + 5 \cdot 3 (\hat{i} \cdot \hat{k}) \]
Since \(\hat{i} \cdot \hat{i} = 1\) and \(\hat{i} \cdot \hat{j} = 0\), \(\hat{i} \cdot \hat{k} = 0\):
\[ W = 5 \cdot 11 \cdot 1 + 0 + 0 = 55 \text{ J} \]

### Final Answer
The work done is:
\[ W = 55 \text{ J} \]

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