This problem requires calculating the work done on a particle when subjected to multiple forces and experiencing a displacement. The work done is determined by the dot product of the net force acting on the particle and the displacement vector.

Calculating Net Force

First, we need to find the total force acting on the particle. This is done by adding the individual force vectors provided:

• Force 1: $\vec{F_1} = 3\hat{i} + 2\hat{j} + 5\hat{k}$
• Force 2: $\vec{F_2} = 2\hat{i} + \hat{j} - 3\hat{k}$

The net force, $\vec{F}_{net}$, is the sum of these forces:

$\vec{F}_{net} = \vec{F_1} + \vec{F_2}$

$\vec{F}_{net} = (3\hat{i} + 2\hat{j} + 5\hat{k}) + (2\hat{i} + \hat{j} - 3\hat{k})$

Combine the components:

$\vec{F}_{net} = (3+2)\hat{i} + (2+1)\hat{j} + (5+(-3))\hat{k}$

$\vec{F}_{net} = 5\hat{i} + 3\hat{j} + 2\hat{k}$

Determining Displacement Vector

Next, we determine the displacement vector ($\vec{d}$) of the particle. This is found by subtracting the initial position vector from the final position vector:

• Initial Position: $\vec{r_1} = 2\hat{i} - \hat{j} - 3\hat{k}$
• Final Position: $\vec{r_2} = 4\hat{i} - 3\hat{j} + 7\hat{k}$

The displacement vector is calculated as:

$\vec{d} = \vec{r_2} - \vec{r_1}$

$\vec{d} = (4\hat{i} - 3\hat{j} + 7\hat{k}) - (2\hat{i} - \hat{j} - 3\hat{k})$

Subtract the components:

$\vec{d} = (4-2)\hat{i} + (-3 - (-1))\hat{j} + (7 - (-3))\hat{k}$

$\vec{d} = 2\hat{i} + (-3 + 1)\hat{j} + (7 + 3)\hat{k}$

$\vec{d} = 2\hat{i} - 2\hat{j} + 10\hat{k}$

Work Done Calculation

The work done ($W$) is the dot product of the net force ($\vec{F}_{net}$) and the displacement vector ($\vec{d}$). The formula is:

$W = \vec{F}_{net} \cdot \vec{d}$

Using the calculated vectors:

$W = (5\hat{i} + 3\hat{j} + 2\hat{k}) \cdot (2\hat{i} - 2\hat{j} + 10\hat{k})$

To compute the dot product, multiply the corresponding components and sum the results:

$W = (5 \times 2) + (3 \times -2) + (2 \times 10)$

$W = 10 + (-6) + 20$

$W = 10 - 6 + 20$

$W = 4 + 20$

$W = 24$

Therefore, the work done by the forces is 24 units.