Lorentz transformation equations between inertial frames of reference moving in the three dimensions of space

The Lorentz transformation equations, which are considered the cornerstone of the special theory of relativity, are fundamental theoretical tools to describe the transformation of spacetime coordinates between frames of reference moving at a constant velocity relative to each other. Here we obtain the Lorentz transformation equations between frames of reference moving in the three dimensions of space and further we present the matrix form of the obtained transformation equations with the help of the position six-vector. Moreover, we discuss the notion of six-velocity and six-momentum, which appear to be unknown in the literature, and we also investigate its main consequences, including demonstrating that it is completely consistent with many of the familiar outcomes of four-vector and deriving the transformation of six-momentum and energy that are completely compatible with the Lorentz invariant energy-momentum relation.