A force `vec(F) = 5i - 3j + 2k` moves a particle from `vec(r )_(1) = 2i + 7j + 4k " to " vec_(r )_(2) = 5i + 2j + 8k`. Calculate the workdone

Options

A. 38 units
B. 20 units
C. 30 units
D. 15 units

Answer

To calculate the work done by the force vector \(\vec{F} = 5\hat{i} - 3\hat{j} + 2\hat{k}\) when moving a particle from position vector \(\vec{r}_1 = 2\hat{i} + 7\hat{j} + 4\hat{k}\) to position vector \(\vec{r}_2 = 5\hat{i} + 2\hat{j} + 8\hat{k}\), we can follow these steps:

### Step 1: Calculate the Displacement Vector
The displacement vector \(\vec{d}\) can be calculated using the formula:
\[ \vec{d} = \vec{r}_2 - \vec{r}_1 \]
Substituting the values:
\[ \vec{d} = (5\hat{i} + 2\hat{j} + 8\hat{k}) - (2\hat{i} + 7\hat{j} + 4\hat{k}) \]

### Step 2: Simplify the Displacement Vector
Now, we simplify the expression:
\[ \vec{d} = (5 - 2)\hat{i} + (2 - 7)\hat{j} + (8 - 4)\hat{k} \]
\[ \vec{d} = 3\hat{i} - 5\hat{j} + 4\hat{k} \]

### Step 3: Calculate the Work Done
The work done \(W\) by the force vector is given by the dot product of the force vector \(\vec{F}\) and the displacement vector \(\vec{d}\):
\[ W = \vec{F} \cdot \vec{d} \]
Substituting the values:
\[ W = (5\hat{i} - 3\hat{j} + 2\hat{k}) \cdot (3\hat{i} - 5\hat{j} + 4\hat{k}) \]

### Step 4: Perform the Dot Product
Calculating the dot product:
\[ W = (5 \cdot 3) + (-3 \cdot -5) + (2 \cdot 4) \]
\[ W = 15 + 15 + 8 \]
\[ W = 38 \]

### Final Answer
The work done by the force in moving the particle from \(\vec{r}_1\) to \(\vec{r}_2\) is:
\[ \boxed{38 \text{ units}} \]