It can be much simpler and much less error prone (user errors) to transform forces and moments between two points by using a cross product instead of using transformation matrices like:
The simple trick is to recognize that a cross product can be expressed as a matrix multiply as follows:
So to show how we would use the cross product to our advantage, see the the following example. We want to transform the forces and moments applied to a rigid body at point A to point B:
It’s quite easy to see that the forces along the cartesian coordinates remain the same between the two points, but the moments change due to the moment arm (ptb-pta).
It is also useful that displacements can be easily transformed using the transpose of the transformation matrix above: