"They" don't want you to divide vectors by vectors, but it's one of the easiest to understand operations you can define. We'd like a scalar q that makes qB = A, but that's not always possible, so pick the value of q that gets as close as possible, minimizing the size of the remainder R = A - qB. This happens when qB is the component of A parallel to B, and R is the component of A perpendicular to B. We'll call q the scalar quotient and write it as A/B.