Three-Dimensional Force Systems

A three-dimensional force system is one whose force lines of action are not confined to a single plane. Its analytical description requires three coordinate axes, vector moments, and the rules of the vector cross product.

Force in three dimensions

In Cartesian coordinates, a force is written as $\vec F=F_x\vec i+F_y\vec j+F_z\vec k$. The three components uniquely define the force vector, and their signs determine the component directions along the coordinate axes.

If a three-dimensional force is specified by its Cartesian components $F_x$, $F_y$, and $F_z$, its magnitude is:

$$F=\sqrt{F_x^2+F_y^2+F_z^2}.$$

The components are algebraic quantities whose signs determine the vector direction, while the force magnitude is always nonnegative.

Direction cosines

If $\alpha$, $\beta$, and $\gamma$ are the angles between the force vector and the positive $x$, $y$, and $z$ axes, then $F_x=F\cos\alpha$, $F_y=F\cos\beta$, and $F_z=F\cos\gamma$. The direction cosines satisfy $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$.

Force along a specified line

If a force is directed from point $A(x_A,y_A,z_A)$ toward point $B(x_B,y_B,z_B)$, first form the vector $\overrightarrow{AB}$. Normalize it to obtain $\vec e_{AB}=\overrightarrow{AB}/|\overrightarrow{AB}|$, then write the force as $\vec F=F\vec e_{AB}$.

Moment of a force about a point

The moment of a force $\vec F$ applied at point $A$ about point $O$ is $\vec M_O=\vec r\times\vec F$, where $\vec r=\overrightarrow{OA}$. The moment vector is perpendicular to the plane formed by $\vec r$ and $\vec F$, with its direction given by the right-hand rule.

Moment of a force about an axis

The moment about a coordinate or arbitrary axis equals the projection of the force-moment vector onto that axis. If $\vec e$ is a unit vector along the axis, then $M_{axis}=\vec e\cdot(\vec r\times\vec F)$.

Resultant force vector and resultant moment

For a three-dimensional force system, the resultant force vector is $\vec R=\sum\vec F_i$, and the resultant moment about point $O$ is $\vec M_O=\sum(\vec r_i\times\vec F_i)+\sum\vec M_j$. In a 3D problem, each of these vectors has three Cartesian components.

Example

Suppose a force has components $F_x=3$ kN, $F_y=4$ kN, and $F_z=12$ kN. Its magnitude is $F=\sqrt{3^2+4^2+12^2}=13$ kN. The force vector can be written as $\vec F=(3\vec i+4\vec j+12\vec k)$ kN.

Common mistakes

• using a planar moment sign convention for a three-dimensional moment vector;
• failing to normalize the vector between two points before multiplying it by a specified force magnitude;
• confusing the moment about a point with its projection onto an axis;
• reversing the order in $\vec r\times\vec F$, which reverses the moment direction;
• omitting one of the three force or moment components.