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** (*dr _{i}/dt_{i}*); 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 *r _{1}* and

*t*like this:

_{1}The velocity vector is thus a three-vector because a particle has three dimensions of movement: **v** = (*v _{1}, v_{2}, v_{3}*) = (

*dr*,

_{1}/dt_{1}*dr*,

_{2}/dt_{2}*dr*). Similarly, the acceleration vector is a three-vector:

_{3}/dt_{3}**a**= (

*a*) = (

_{1}, a_{2}, a_{3}*d²v*). These may be represented by component-wise vector division:

_{1}/dt_{1}², d²v_{2}/dt_{2}², d²v_{3}/dt_{3}²**v**=

*d*

**r**/

*d*

**t**and

**a**=

*d*

**v**/

*d*

**t**.

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 (*r _{b} – r_{a}*) / (

*t*) 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: (

_{b}– t_{a}*t*) / (

_{b}– t_{a}*r*) for which the spatial component is the independent variable.

_{b}– r_{a}