**Detailed Explanation**
**1. What is the net force acting on the knot in terms of F1, F2, and F3?**
Since the lantern is in equilibrium, the net force acting on the knot must be zero. This means that the forces exerted by the three strings must balance each other out. We can express this mathematically as follows:
```
F_net = F_1 + F_2 + F_3 = 0
```
**2. Find the magnitudes of the x and y components for each force acting on the knot.**
To do this, we need to break down each force into its x and y components. We can use the following equations:
```
F_x = F * cos(θ)
F_y = F * sin(θ)
```
where θ is the angle between the force and the x-axis.
For string 1, we have:
```
F_{1x} = F_1 * cos(θ_1)
F_{1y} = F_1 * sin(θ_1)
```
For string 2, we have:
```
F_{2x} = F_2 * cos(θ_2)
F_{2y} = F_2 * sin(θ_2)
```
For string 3, we have:
```
F_{3x} = F_3 * cos(θ_3)
F_{3y} = F_3 * sin(θ_3)
```
**3. What is the magnitude of the net force acting on the knot in the x direction and in the y direction?**
To find the net force in the x direction, we simply sum the x components of all the forces:
```
F_{xnct} = F_{1x} + F_{2x} + F_{3x}
```
To find the net force in the y direction, we simply sum the y components of all the forces:
```
F_{ynct} = F_{1y} + F_{2y} + F_{3y}
```
**4. Assume that θ1 = 30° , θ2 = 60° and the mass of the lantern is 2.1 kg. Find F1, F2, and F3.**
To find the magnitudes of the forces F1, F2, and F3, we can use the following equations:
```
F_1 = mg / sin(θ_1)
F_2 = mg / sin(θ_2)
F_3 = mg / sin(θ_3)
```
where mg is the weight of the lantern.
Substituting the given values, we get:
```
F_1 = (2.1 kg)(9.81 m/s^2) / sin(30°) = 42.7 N
F_2 = (2.1 kg)(9.81 m/s^2) / sin(60°) = 21.3 N
F_3 = (2.1 kg)(9.81 m/s^2) / sin(90°) = 21.3 N
```
**Conclusion**
The forces exerted by the three strings on the knot are:
* F1 = 42.7 N
* F2 = 21.3 N
* F3 = 21.3 N
The net force acting on the knot in the x direction is zero. The net force acting on the knot in the y direction is also zero.