The traditional ratio, x : y, represents both x / y and y / x. In order to represent a ratio as quotients, both forms are required. Let us define a ratio as an ordered pair of quotients:

For vectors this means

One can either exclude zero or include infinity as follows;

A rate of change is

A vector rate of change is

A point rate of change (speed, pace) is

A vector point rate of change (velocity, lenticity) is

A second order point rate of change (acceleration, relentation) is