A particle moved from position `vec r_(1) = 3 hat i + 2 hat j - 6 hat k` to position `vec r_(2) = 14 hat i + 13 hat j + 9 hat k ` under the action of a force `( 4 hat i + hat j + 3 hat k)` newton. Find the work done

Options

A. 10 J
B. 100 J
C. 0.01 J
D. 1 J

Answer

To find the work done by the force as the particle moves from one position to another, we can follow these steps:

### Step 1: Identify the initial and final position vectors
The initial position vector is given as:
\[ \vec{r_1} = 3 \hat{i} + 2 \hat{j} - 6 \hat{k} \]
The final position vector is given as:
\[ \vec{r_2} = 14 \hat{i} + 13 \hat{j} + 9 \hat{k} \]

### Step 2: Calculate the displacement vector
The displacement vector \(\vec{d}\) can be calculated as:
\[ \vec{d} = \vec{r_2} - \vec{r_1} \]
Substituting the values:
\[ \vec{d} = (14 \hat{i} + 13 \hat{j} + 9 \hat{k}) - (3 \hat{i} + 2 \hat{j} - 6 \hat{k}) \]
Now, perform the subtraction component-wise:
\[ \vec{d} = (14 - 3) \hat{i} + (13 - 2) \hat{j} + (9 + 6) \hat{k} \]
\[ \vec{d} = 11 \hat{i} + 11 \hat{j} + 15 \hat{k} \]

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

### 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:
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the vectors:
\[ W = (4 \hat{i} + 1 \hat{j} + 3 \hat{k}) \cdot (11 \hat{i} + 11 \hat{j} + 15 \hat{k}) \]
Calculating the dot product:
\[ W = (4 \cdot 11) + (1 \cdot 11) + (3 \cdot 15) \]
\[ W = 44 + 11 + 45 \]
\[ W = 100 \text{ joules} \]

### Final Answer
The work done by the force is:
\[ \boxed{100 \text{ joules}} \]