As we have seen in Chapter 3, it is possible to introduce operations with vectors having formal properties very similar to the formal properties which characterize the ordinary algebraic operations with numbers. One may wonder whether or not one can actually reduce vector operations to ordinary algebraic operations. This is indeed possible. In fact, the reduction of geometrical operations to operations with numbers, initiated by Descartes, has proved to be a decisive step—if not the most decisive one—in the development of geometry. This general reduction has made it possible to reduce all specific computational work in geometry to computations with numbers and to derive geometrical theorems from theorems on operations with numbers. Moreover, this reduction has made it easy to free geometry from the restriction to our three dimensional space and to extend the concepts of geometry to so-called spaces of more than three dimensions, in fact to spaces of infinitely many dimensions. In all this work vectors are the most effective tool.

The notion of vector and the rules of operations with vectors were developed during the second half of the nineteenth century in a rather roundabout way. The proper understanding of vectors and vector operations was attained about 1880 through the work of J. Willard Gibbs of Yale.

In the following discussion—just for the sake of convenience—we shall restrict ourselves to the plane and assume all vectors to lie in this plane.