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.

 

[CellCoree]

The statement in your book applies to any pair of planes, not just parallel planes. The value of d in the equation represents the distance of the plane from the origin in the direction of the normal vector (a, b, c). Therefore, two planes with the same sign for d will be on the same side of the origin, while two planes with different signs for d will be on opposite sides of the origin. This is true for any pair of planes, whether they are parallel or not. Parallel planes are simply a special case where the normal vectors are parallel, but the same principle applies to all planes.