## Vectors and Vector Spaces

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 ÃŸ, executed one after another, is a third "total" displacement ?. If ÃŸ is applied after by placing the origin of the arrow ÃŸ at the terminus of a, then the combined displacement ? = a + ÃŸ is the arrow leading from the origin of a to the terminus of ÃŸ. This is the diagonal of the parallelogram with sides a and ÃŸ. This rule for finding a + ÃŸ is the so-called parallelogram law for the addition of vectors.