(they know that they need three equations to find a unique solution) --- one for the force balance in the x-direction and the other two for the moment balance about two different points.


Is this a trick question?
How can all forces be 0?

You raise an important point, and I see why it might seem like a trick question. Let me explain.

### Key Insight: Physical Interpretation of the Problem

In statics, a common practice is to use **force balance equations** and **moment balance equations** to find the unknown forces in a system.

- **Force balance** in the \( x \)-direction ensures that the sum of all forces in the \( x \)-direction is zero.
- **Moment balance** about two different points ensures that the system is in rotational equilibrium, meaning that the sum of the moments (torques) about these points is zero.

Given that the system is set up with three equations, one for force balance and two for moment balance, this should allow for the determination of the three unknown forces \( F_1 \), \( F_2 \), and \( F_3 \).

### The Issue: Homogeneous System

However, the system provided in the question is a **homogeneous system** of the form:
\[
A \mathbf{F} = 0
\]
where \( A \) is the coefficient matrix and \( \mathbf{F} \) is the vector of forces \( [F_1, F_2, F_3]^T \). A homogeneous system only has the **trivial solution** (where all forces are zero) if the determinant of \( A \) is non-zero, which is indeed the case here.

This means the only mathematically consistent solution to the equations provided is:
\[
F_1 = F_2 = F_3 = 0
\]
This may seem counterintuitive from a physical perspective because **forces** being **zero** implies there are no forces acting on the system. However, that solution mathematically satisfies the equations.

### Is It a Trick Question?

It could be viewed as a "trick question" because:
- The mathematical formulation of the problem leads to the trivial solution where all forces are zero.
- From a physical standpoint, having all forces equal to zero may not seem realistic if you're expecting to balance actual forces, but the equations themselves do not provide any non-zero solution.

So, **yes**, this might be considered a trick question. The student may expect non-zero forces as a solution, but based on the equations provided, the only valid solution is \( F_1 = F_2 = F_3 = 0 \).

### Physical Context Missing?
The confusion might arise because the problem is purely mathematical without giving context to specific physical forces. In real-world applications, additional conditions (e.g., external loads or constraints) would typically be given to ensure non-zero forces are balanced, leading to a more meaningful solution.

### Conclusion
The equations as given mathematically lead to all forces being zero. If the student is looking for a "unique" non-trivial solution (where forces are non-zero), that wouldn't be possible with this set of equations.

Would you like more discussion on how different physical conditions might alter the problem?

Here are some follow-up questions:
1. What conditions might cause a system to have non-zero solutions?
2. How does the presence of external forces influence the solution?
3. Can a statics problem ever have all forces be zero in a real-world scenario?
4. What happens if we add additional constraints or forces to the system?
5. How does one handle overdetermined or underdetermined systems in statics?

**Tip:** In statics problems, always interpret the results in the context of the physical system. Sometimes, the mathematical solution might imply a situation that doesn't align with physical expectations.

Explore the concept of force and moment equilibrium in statics with a focus on homogeneous systems. Learn how to determine unique solutions using three equations: one for force balance and two for moment balance about different points.

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