Force Vectors and Moments

With the vector notation and computational tools established in the Preliminaries, we are ready to model the most fundamental quantity in mechanics: force. A force is a vector, fully characterized by its magnitude, direction and point of application. When several forces act on a body, their combined effect is captured by the resultant, obtained through vector addition. This simple principle extends naturally from two-dimensional problems with a handful of forces to three-dimensional systems involving rotated coordinate frames and multiple reference points.

We begin with worked examples in two dimensions, progressing from straightforward resultant calculations to problems that require force decomposition along specified directions. We then move into three dimensions, where the main challenge shifts from the forces themselves to the kinematics of describing positions and orientations in space. Rotation matrices, coordinate frame transformations and the right-hand rule become essential tools as soon as we leave the plane.

The part concludes with moments. A moment measures the rotational effect of a force about a point, computed as the cross product of a position vector and a force vector. We develop this concept from simple planar cases, where moments reduce to signed scalars, to the general three-dimensional setting, where moment vectors carry both magnitude and direction. Throughout, we rely on SymPy to handle the cross products, simplifications and substitutions, keeping the focus on modelling rather than arithmetic.