The curve Ax+By=0 is the line through the origin perpendicular to [A,B].

Knowing this we have the properties.
 [A,B].[x,y]=0
 [A,B] perpendicular to [x,y]

This means that all the lines of Ax+By=C are perpendicular to the normal vector.

The line Ax+By=0 is perpendicular to [A,B] and at signed distance C/(A^{2}+B^{2})^{1/2} from the line Ax+By=0.

The signed distance with c>0 are the lines in the direction of [A,B].
The signed distance with c=0 is the line going throught the origin.
The signed distance with c<0 are the lines going in the opposite direction of [A,B].

If we use the line Ax+By=C then we have this picture.

The signed distance between Ax+By=C ' and Ax+By=C '' is
(C ''C ')/(A^{2}+B^{2})^{1/2}

In 3D the equation is similiar. It is(C ''C ')/(A^{2}+B^{2}+C^{2})^{1/2}.
The equations (C ''C ')/(A^{2}+B^{2})^{1/2}and (C ''C ')/(A^{2}+B^{2}+C^{2})^{1/2}
will be useful in discussing the concepts of waves in 2D and 3D. 