• From elementary physics we know vectors as being arrows, exhibiting direction and length. This means that they have both a head as well as a tail, the latter being drawn as a point of the same space in which the physics is enacted. A vector, then, is equivalent to an ordered pair of points in the space. Such a conception works perfectly on the common plane as well as in three-dimensional (Euclidean) space.

However, in general this idea presents difficulties. One can already perceive them clearly on “curved” two-dimensional surfaces (consider, as an example, such a “vector” on a sphere S2 in the case when its length equals the length of the equator). Recall, however, the various contexts in which vectors enter the physics. One comes to the conclusion that the “tail” point of the vector has no “invariant” meaning; only the head point of the vector makes sense as a point of the space. Take as a model case the concept of the (instantaneous) velocity vector v of a point mass at some definite instant of time t. Its meaning is as follows: if the point is at position r at time t, then it will be at position r + εv at time t + ∊. However long the vector v is, the point mass will be only infinitesimally remote from its original position. The (instantaneous) velocity vector v thus evidently carries only “local” information and it is related in no reasonable way to any “tail” point at finite distance from its head.