iSoul In the beginning is reality

Dimensions of movement

The instantaneous movement of a particle may be represented by a velocity vector, which describes an instantaneous motion by its magnitude and direction: the magnitude is the ratio of the differentials of the distance and time in each dimension of the movement (dri/dti); the direction is the direction of the instantaneous tangent in each dimension of the movement. The bold point needs to be emphasized because this is what has been missed. Space and time measurements concern the same movement and have the same dimensions as the movement.

Each axis of movement may be projected onto two-dimensions for visualization, with for example axes r1 and t1 like this:

Space-time axes

The velocity vector is thus a three-vector because a particle has three dimensions of movement: v = (v1, v2, v3) = (dr1/dt1, dr2/dt2, dr3/dt3). Similarly, the acceleration vector is a three-vector: a = (a1, a2, a3) = (d²v1/dt1², d²v2/dt2², d²v3/dt3²). These may be represented by component-wise vector division: v = dr/dt and a = dv/dt.

Note that velocity and acceleration may be defined by component-wise division because the denominator of each component is the independent variable and so can and must be non-zero. For example, in the velocity (rb – ra) / (tb – ta) the denominator is the independent variable, which must be non-zero, and the numerator is the dependent variable, which can be any real number. For the celerity it is the opposite: (tb – ta) / (rb – ra) for which the spatial component is the independent variable.

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