We propose a novel approach to the teaching of undergraduate planar mechanism dynamics. To illustrate the approach, we use a case study, the dynamics of the planar slider-crank mechanism. In this case study, we make extensive use of an operator representing in two-dimensional form the cross-product of two vectors. Furthermore, by using the natural orthogonal complement, introduced elsewhere, we produce a systematic procedure to derive a dynamic model of the same class of mechanism. Subsequently, we illustrate how, with the use of the aforementioned operator, the dynamic balancing of this mechanism, as first proposed by Berkof and Lowen for RRRR planar linkages, and extended by Bagci to the slider-crank mechanism, simplifies tremendously.