The polar coordinate system is an effective way of representing the positions of bodies with the angle they make with the origin, and the distance they are away from it. Polar coordinates are useful for dealing with motion around a central point -- just the case we have with planets moving around the Sun.

However, to continue with our use of vectors, we must define a few polar basis vectors. Our first defined vector will be the unit radial vector. It will be represented by r and will be defined as

This is a fortunate definition; according to it,

Since we have two Cartesian basis vectors, i and j, we should also have two polar basis vectors. The second basis vector, which we shall call the unit transverse vector and represent with theta, is defined as the rate of change of r with respect to theta:

Note that if we again take the derivative of theta with respect to theta we find that

Let us, for the sake of an example, see what our velocity vector v and our angular momentum vector l would look like in terms of this new polar system. (We shall require them later in the proof anyway.)

Velocity is the instantaneous rate of change of the position of the planet: