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### Problem 19:
Forces \( F_1 = 1 \text{ kN}, F_2 = 3 \text{ kN}, F_3 = 2 \text{ kN}, F_4 = 5 \text{ kN}, F_5 = 2 \text{ kN} \) act along the lines joining the corners of a parallel piped whose sides are \( 2.5 \text{ m}, 2 \text{ m}, \text{ and } 1.5 \text{ m} \), as shown in the diagram. The coordinates for points \( O, A, B, C, D, E, F, G \) are given as points in space.

The goal is to:
1. Find the **resultant force**.
2. Find the **moment of the force couple** at the origin \( O \).

### Solution:
The given problem involves the following steps:
- **Coordinates of points** \( O(0,0,0), A(2.5,0,0), B(2.5,2,0), C(0,2,0), D(0,0,1.5), E(2.5,0,1.5), F(2.5,2,1.5), G(0,2,1.5) \).
- **Force vectors** at the specific points were computed by determining the unit vector direction and multiplying it with the force magnitudes.
- The **resultant force** is found by vector summation of all the forces.
- The **moment of the force couple** is calculated using the position vectors relative to the origin and their corresponding forces, using the cross-product formula.

The steps involve:
1. Calculation of each **force vector** \( \mathbf{F}_i \).
2. Summing all forces to get the **resultant force** \( \mathbf{F}_{res} \).
3. Using the formula \( \mathbf{M}_O = \sum (\mathbf{r}_i \times \mathbf{F}_i) \) to calculate the **moment** at the origin.

Would you like a detailed breakdown of these calculations, or assistance in a specific section of this problem?

Here are some questions to deepen your understanding:
1. What is the significance of finding the resultant force?
2. How is the moment of a force calculated using cross-products?
3. Why do we use unit vectors when calculating force directions?
4. What is the difference between the force couple and the resultant force?
5. How would the problem change if the forces acted at different points?

**Tip**: Always double-check the units when working with forces and moments, as unit consistency is key in vector problems.

Forces F1 = 1 kN, F2 = 3 kN, F3 = 2 kN, F4 = 5 kN, F5 = 2 kN act along the lines joining the corners of a parallel piped whose sides are 2.5 m, 2 m, and 1.5 m. Find the resultant force and the moment of the force couple at the origin O.

This problem involves calculating the resultant force and moment of a force couple at the origin O, with forces acting along the lines joining the corners of a parallel piped. The solution applies vector summation and cross-product methods for determining the moment of forces. Ideal for college-level students studying engineering or physics.

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