The proof that there are three time dimensions is based on showing that a temporal position is a vector, that is, it has magnitude and direction. That may be shown by considering three orthogonal movements of an object. Let the position of the object be represented by a point (as on a corner) relative to an origin point.
Consider the travel times of three separate orthogonal movements from their distances traveled divided by their average speeds, times their unit vectors (e1, e2, and e3):
(s1 / v1) e1, then (s2 / v2) e2, then (s3 / v3) e3.
The resultant duration vector t is found by adding each of the three orthogonal vectors together by vector addition:
t = (s1 / v1) e1 + (s2 / v2) e2 + (s3 / v3) e3.
That is, the resultant duration vector t has three components:
t = t1 e1 + t2 e2 + t3 e3.
Since t1, t2, and t3 can be different, they represent three different components of the vector t. That is, t1, t2, and t3 are orthogonal components of a temporal vector t:
t = (t1, t2, t3).
Thus we have demonstrated three dimensions of time.
These three dimensions are based on the same directions as displacement (spatial), velocity, force, and other physical vectors.