A force of `vecF=2xhati+2hatj+3z^2hatk N` is acting on a particle. Find the work done by this force in displacing the body from `(1, 2, 3)m` to `(3, 6, 1)m`.

Options

A. (a) `-10J`
B. (b) `100J`
C. (c) `10J`
D. (d) `1J`

Explanation

To find the work done by the force \(\vec{F} = 2x \hat{i} + 2 \hat{j} + 3z^2 \hat{k}\) when displacing the particle from the point \((1, 2, 3)\) to \((3, 6, 1)\), we will follow these steps:

### Step 1: Identify the displacement vector
The displacement vector \(\vec{d}\) can be calculated as:
\[ \vec{d} = (x_f - x_i) \hat{i} + (y_f - y_i) \hat{j} + (z_f - z_i) \hat{k} \]
where \((x_i, y_i, z_i) = (1, 2, 3)\) and \((x_f, y_f, z_f) = (3, 6, 1)\).

Calculating the components:
\[ \vec{d} = (3 - 1) \hat{i} + (6 - 2) \hat{j} + (1 - 3) \hat{k} = 2 \hat{i} + 4 \hat{j} - 2 \hat{k} \]

### Step 2: Write the work done expression
The work done \(W\) by the force during the displacement can be expressed as:
\[ W = \int_C \vec{F} \cdot d\vec{r} \]
where \(d\vec{r} = dx \hat{i} + dy \hat{j} + dz \hat{k}\).

### Step 3: Substitute the force and displacement
The force \(\vec{F}\) varies with position, so we will express the work done in terms of the coordinates:
\[ W = \int_{(1, 2, 3)}^{(3, 6, 1)} (2x \, dx + 2 \, dy + 3z^2 \, dz) \]

### Step 4: Set up the limits for integration
We will integrate each component separately, with the limits for \(x\) from 1 to 3, for \(y\) from 2 to 6, and for \(z\) from 3 to 1.

### Step 5: Calculate the work done
1. **Integrate with respect to \(x\)**:
\[ W_x = \int_{1}^{3} 2x \, dx = [x^2]_{1}^{3} = 3^2 - 1^2 = 9 - 1 = 8 \]

2. **Integrate with respect to \(y\)**:
\[ W_y = \int_{2}^{6} 2 \, dy = [2y]_{2}^{6} = 2(6) - 2(2) = 12 - 4 = 8 \]

3. **Integrate with respect to \(z\)**:
\[ W_z = \int_{3}^{1} 3z^2 \, dz = -\int_{1}^{3} 3z^2 \, dz = -[z^3]_{1}^{3} = -[27 - 1] = -26 \]

### Step 6: Combine the results
Now, we sum the contributions from each component:
\[ W = W_x + W_y + W_z = 8 + 8 - 26 = -10 \]

### Final Answer
The work done by the force in displacing the body from \((1, 2, 3)\) to \((3, 6, 1)\) is:
\[ \boxed{-10 \, \text{J}} \]