A vector in the Cartesian plane is an ordered pair (a, b) of real numbers. This is the mathematician's concise definition of a two-dimensional vector. Physicists and engineers like to develop this concept a bit more for the purpose of applying vectors in their disciplines. Thus, they like to think of the mathematician's ordered pairs as representing displacements, velocities, accelerations, forces, and the like. Since such things have magnitude and direction, they like to imagine vectors as arrows in the plane whose magnitudes are their lengths and whose directions are the directions that the arrows are pointing. The two small dark arrows shown in part (a) of the drawing below form our point of departure in the study of vectors in the plane.

These are called the unit basis vectors, or unit vectors. From them all other vectors arise. To the mathematician, they are simply the ordered pairs...