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Forces in Space

In the previous chapter, we focused our discussion on coplanar force systems—that is, forces in two dimensions. In this chapter, we extend our discussion to forces in three dimensions.

The figure below shows a force F in a three-dimensional system. The components of the force are proportional to the components of its distance from tail to head. In equation form, it is expressed as:

$$ \frac{F_x}{F} = \frac{x}{L} \quad \text{(1)} $$
$$ \frac{F_y}{F} = \frac{y}{L} \quad \text{(2)} $$
$$ \frac{F_z}{F} = \frac{z}{L} \quad \text{(3)} $$
Concept

Example 1: A force F = 340 kN has its tail at the origin and its head at the point (3, 5, 7).

Distance from Tail to Head:

$$ L = \sqrt{x^2 + y^2 + z^2} = \sqrt{3^2 + 5^2 + 7^2} = 9.11 $$

Component Calculations:

$$ F_x = \frac{3}{9.11} \cdot 340 = 111.96 \text{ kN} $$
$$ F_y = \frac{5}{9.11} \cdot 340 = 186.61 \text{ kN} $$
$$ F_z = \frac{7}{9.11} \cdot 340 = 261.25 \text{ kN} $$

Sign Convention:

Proper sign convention should be considered. Oppositely directed forces have opposite signs. The right-handed sign convention is commonly used:

Moment of a Force in Three–Dimensional System

We define a moment as the product of the magnitude of the force and the perpendicular distance from the line of action of the force to the point:

$$ M = F \cdot d $$

where \( d \) is the moment arm, the perpendicular distance.

The principle of moments still applies in three-dimensional force systems, and is expressed as:

$$ M_A = \sum \left( \mathbf{F} \times \mathbf{d} \right) $$

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