We develop a dynamics on a vector bundle that accurately describes the mechanical behavior of a directed medium. The directed medium is a continuum with microstructures, which is described by a deformable vector, called a director. In geometric continuum mechanics, an elastic body is viewed as a differentiable manifold, while a directed medium is viewed as a vector bundle whose fiber denotes a collection of the deformable directors. In this study, we begin with geometrical settings of the continuum dynamics on a tangent bundle of a vector bundle, and derive a weak form and equations of motion for the directed medium. Moreover, we apply our resultant equations to a Cosserat rod, as an example, and find that the derived equations of motion coincide with the balance laws of large deformable rods. Additionally, the equations of motion for the Cosserat rod are reduced to those for the special case of the Cosserat rod under undeformed cross-sectional conditions. Finally, we discuss an application of our Cosserat rod results to biopolymers and comment on the analyses of their smaller microstructures. This is important for future development because the microstructural changes of the biopolymers are deeply related to their macro conformations and, eventually, to the biological functions depending on both the macro and micro conformations.