I think I can.

First, the way I would do it. You have three points and all lying on the plane. If you create two vectors connecting two of the three points, say and . If you cross these, this will give the normal to the plane

.

Suppose that is a point lying on the plane then the vector throught this point and is

(every line on the plane will contain this vector)

Now you want every line on the plane perpendicular to the normal so

What I think the author is doing.

If you associate vectors for each point P_0, P_1 and P_2, from the origin to the point and denote these by and . Now

and

so

.

Now associate a vector from origin to as denote this as . Thus, the vector from to is

.

Now dot these two

so you see the answers are the same.