Abstract: Spacetime geometric algebra is a unified mathematical language for physics. The geometric representation of spacetime and the use of four-vectors are vital to the successful findings of three-dimensional, four-dimensional non-Euclidean geometry in Lorentz and Galilean transformations. Thus, the usual opinion that there is a sole set of events presents now in a three-dimensional or four-dimensional spacetime cannot be continual. The geometric representations discussed in this study include the following: Minkowski’s path to spacetime Galilean transformation as a geometrical representation of motion Lorentz transformation as a geometrical representation of inertial frames and worldline of a particle. The expansion of the special relativity theory using four-vectors space-time four-vectors.