Can anyone give me a good explanation of why the plane Ax+By+Cz=D has normal vector (A,B,C)? It just seems so random.
What you must understand is that the normal is perpendicular to ever line contained in the plane.
That is the dot product of the normal with any vector determined by two points in the plane is equal to zero.
The usual form of a plane is : where is a point in the plane and the normal is .