The canonical coordinate systems (rectangular, polar and spherical) are sometimes not the best for studying the trajectories of some forms of motions. For example, motion of objects in an elliptical orbit being described by polar or spherical coordinates may not be accurate. It is due to this that we have derived the position vectors, velocity vectors, acceleration vectors, simple representation of magnitude of the velocity and equations of motion in the elliptical coordinate system. An attempt was also made towards solving the derived equations of motion. The general algorithm for conversion among coordinate systems was also provided.