In Chapter 4, Shirley provides the equation used to check whether a point lies on the surface of a sphere in a 3-dimensional Euclidian space. It has to hold for

x*x + y*y + z*z = r*r

Here, r is the radius of the sphere and (x, y, z) represents an arbitrary point - the previous wording is just another interpretation of the equation for a sphere S centered at the origin of radius r.

He then reminds us of the equation for the vector that points from a point C to a point p - in our case, C is the center of the sphere and p is a point on the surface of the sphere:

v = p - C

We know that the square of a vector v = (x, y, z) is the dot-product of the vector multiplied with itself, which ultimately is the square of its length, since

v . v = (x, y, z) . (x, y, z) = x*x + y*y + z*z = |v| * |v| 

(. denotes the operator for the dot-product here)

Given the equation for a sphere centered at the origin C we can thus conclude:

(p - C) . (p - C) = v . v = |v| * |v| = r * r