The problem involves a structure that is supported by a ball-and-socket joint at point $A$ and two cables connected at points $B$ and $C$. Additionally, there is a uniformly distributed load of $800 \, \text{N/m}$ applied along part of the structure.

Steps for solving:

1. Reactions at A: Since the ball-and-socket joint allows rotation in all directions but not translation, there will be three reaction forces at point $A$ (in the $x$-, $y$-, and $z$-directions). Let's call these reactions $A_x$, $A_y$, and $A_z$.

2. Tensions in Cables: The cables $DB$ and $DC$ will experience tension forces. The tension in each cable acts along the line of the cable, so we need to express the forces $T_{DB}$ and $T_{DC}$ in vector form.

3. Force Equilibrium: For the structure to be in equilibrium, the sum of the forces in the $x$-, $y$-, and $z$-directions must be zero: $\sum F_x = 0, \quad \sum F_y = 0, \quad \sum F_z = 0$

4. Moment Equilibrium: Similarly, the sum of the moments about point $A$ must be zero. We will take the moments about point $A$ in all three directions to ensure rotational equilibrium: $\sum M_x = 0, \quad \sum M_y = 0, \quad \sum M_z = 0$

Steps to proceed:

1. Determine the geometry:

   • Point $A$ is at the origin (0, 0, 0).
   • Point $B$ is at $(1.5, 1.5, 1.5)$ m.
   • Point $C$ is at $(3, 0, 1.5)$ m.
   • Point $D$ is at $(1, 0, 0)$ m.

2. Determine the vectors for cables:

   • The direction of the force in cable $DB$ is along the vector from $D$ to $B$, and similarly for cable $DC$.
   • The tension forces can then be expressed in terms of their components using these vectors.

3. Apply equilibrium equations: Using the force and moment equilibrium equations, you can solve for the unknowns: the reaction components at $A$ and the tensions in cables $DB$ and $DC$.

Would you like a detailed step-by-step calculation of the reactions at $A$ and the tensions in the cables? Let me know if you'd like specific assistance on any part.

Follow-up Questions:

1. What is the exact force acting at each cable connection point?
2. How do you express the direction of the tension forces in vector form?
3. What are the moment equilibrium equations at point $A$?
4. How does the distributed load affect the reaction forces at $A$?
5. What methods can be used to solve for multiple unknowns in the system of equations?

Tip: When working with vector forces in 3D, it's crucial to break down all forces and moments into their components in each direction (x, y, z) for accurate calculations.