Parallel Planes: Same/Opposite Sides of Origin?

[PFuser1232]

According to my book, if we write the equation of a plane as:

$ax + by + cz = d$

And two planes have values of $d$ with the same sign, they are on the same side of the origin. If they have values of $d$ with different signs, they are on opposite sides of the origin.

I'm confused as to whether this applies strictly to two parallel planes, or any pair of planes.

 

[jbriggs444]

You must decide what "same side of the origin" means for non-parallel planes.

 

[PFuser1232]

You must decide what "same side of the origin" means for non-parallel planes.

What about parallel planes? If both sides of the plane equation (cartesian form) are multiplied by -1, the position of the plane is unaltered, right?

 

[jbriggs444]

Right. In order to make the "parallel plane" version work, you need for a, b and c to either remain constant or at least to remain proportional and not change signs.