chapter  7
46 Pages

Vectors and Vector Spaces

WithGarrett Birkhoff, Saunders Mac Lane

In physics there arise quantities called vectors which are not merely numbers, but which have direction as well as magnitude. Thus a parallel displacement in the plane depends for its effect not only on the distance but also on the direction of displacement. It may conveniently be represented by an arrow a of the proper length and direction. The combined effect of two such displacements a and ß, executed one after another, is a third "total" displacement ?. If ß is applied after by placing the origin of the arrow ß at the terminus of a, then the combined displacement ? = a + ß is the arrow leading from the origin of a to the terminus of ß. This is the diagonal of the parallelogram with sides a and ß. This rule for finding a + ß is the so-called parallelogram law for the addition of vectors.