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We are tasked with calculating the net force \(\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3\) and finding an additional force \(\vec{G}\) such that equilibrium occurs, i.e., \(\vec{F}_{\text{net}} + \vec{G} = 0\).

### Step 1: Summing the forces
Each force is given in component form:

- \(\vec{F}_1 = (-3, -1)\)
- \(\vec{F}_2 = (0, -3)\)
- \(\vec{F}_3 = (3, 4)\)

Let's calculate the net force by summing the components.

\[
\vec{F}_{\text{net}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 = (-3, -1) + (0, -3) + (3, 4)
\]

\[
\vec{F}_{\text{net}} = (-3 + 0 + 3, -1 + (-3) + 4)
\]
\[
\vec{F}_{\text{net}} = (0, 0)
\]

### Step 2: Equilibrium condition
Since \(\vec{F}_{\text{net}} = (0, 0)\), the forces are already in equilibrium. Therefore, the additional force \(\vec{G}\) required for equilibrium is:

\[
\vec{G} = (0, 0)
\]

### Conclusion
The net force is \(\vec{F}_{\text{net}} = (0, 0)\), and no additional force is needed for equilibrium.

Would you like more details on vector addition or equilibrium? Here are some related questions for further practice:

1. What is the formula for vector addition in two dimensions?
2. How is equilibrium defined in physics?
3. What role do forces play in determining equilibrium?
4. How can we calculate the magnitude of a vector from its components?
5. What is the difference between static and dynamic equilibrium?

**Tip:** When calculating the net force, always sum the components in the x and y directions separately for clarity and accuracy.

Dadas las siguientes fuerzas, calcule la fuerza neta (F1 + F2 + F3) y encuentre una fuerza adicional G tal que ocurra equilibrio (si F_Neta = 0, se dice que las fuerzas están en equilibrio). F1 = (-3, -1), F2 = (0, -3), F3 = (3, 4).

Learn how to calculate the net force by summing vectors and finding an additional force G to maintain equilibrium. This example involves vector components and equilibrium conditions. Suitable for students in Grades 9-12.

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