1. If you have the equation of a plane in the form:
then you know that the normal vector of the plane is with the length
If you calculate then you get a vector with the same direction as with the length 1. (= unit vector)
2. Divide the equation of the plane by the length of the normal vector:
The summand is the distance of the origin to the plane.
Therefore at i) the distance of the origin to the plane is
3. If you have a point P with it's stationary vector then the dotproduct is the vertical projection of on . If you now subtract the distance of the origin to the plane from the dotproduct you'll get the distance of the point P to the plane. Because you have to subtract the distance of the origin to the plane it is absolutely necessary that the constant of the equation of the plane is negative.
4. If the point and the origin are on different sides of the plane then the distance of to the plane is positive.
If and the origin are on the same side of the plane then the distance of to the plane is negative.
5. So don't use the absolute values when calculating the distance of a point to a plane and you can easily state on which sides of the plane the point and the origin are.