An Axiomatic Basis for Vectors

Extract

Vector geometry is often developed as a subsection of a geometrical structure, much of which is unrelated to the vector properties deduced. The basic results depend on the law which may be stated in the form AB + BC = AC. This requires only that the equality of two vectors is meaningful, and that if AB = XY and BC = YZ, then AC = XZ (if “equals” are added to “equals” the results are equal). We need not assume that A, B, C,… represent points—they could be for example lines in a plane, planes in space or even elements of a group. A vector may be thought of as a “direction” for changing from one element to another: thus AB = CD means that the way of changing from C to D is essentially the same as the way of changing from A to B. We develop a structure which has these properties.