How does hyperplane work?

Question

The definition of a plane in $R^3$ is intuitive : $Ax + By + Cz = D$ , where $(A, B, C)$ is a normal vector to the plane and $D$ is some bias.
I can visualize it dividing the space into two halves (because I'm a 3-dimensional creature and it's quite intuitive for me) , but how does the generalization of this equation to higher dimensions work?

How do hyperplanes divide space into two halves? ANY explanation is appreciated.

Answer 1

Hyper PLane :

The most intuitive & simple way is to consider it like this :

$Ax + By + Cz - D = 0$

Let $M_2(x,y) = Ax + By - D $
Let $M_3(x,y,z) = Ax + By + Cz - D $
More generally , let $M_n(x,y,\cdots,z) = Ax + By + \cdots + Cz - D $

When we plug in the Co-ordinates of a Point in $R^2$ , $R^3$ , $R^n$ ,
then we will get the $M$ value which will be either Positive or Zero or Negative ,
there is no other Possibility !!!!

When $M$ is Zero , then it is "on" the hyper-plane.
When $M$ is Positive , then it is "on one side" of the hyper-plane.
When $M$ is Zero , then it is "on the other side" of the hyper-plane.
Thus we have 2 "halves" when we check the Sign of $M$.

When we take $D=0$ , then we can see that the Zero vector will give Zero $M$ automatically ,
hence it means those hyper-planes are going though the Origin.

Hyper Surface :

Same Intuition applies for non-linear cases too , where we will get hyper surfaces.
When we have $x^2+y^2-2xy+100=0$ , we can make it $M(x,y)=x^2+y^2-2xy+100$
We will have 2 "halves" when we check the Sign of $M$ & we get a Curve here.

When we have $R^n$ Case $M(x_1,x_2,\cdots,x_n)=x_1^2+x_2^3 + \cdots +2x_1x_2x_n-D$ , we will have 2 "halves" when we check the Sign of $M$ & we get Hyper Surfaces here.

In all Cases , $M=0$ is the Dividing Value [ the Hyper Plane or the Hyper Surface ] between the two "halves" of $R^n$.

Answer 2

Perhaps the simplest case will help.

In two dimensions a hyperplane is a line. The easiest line to think about in the usual coordinate system is the $x$ axis. That clearly divides the plane into the two regions above and below: $y> 0$ and $y < 0$.

In three dimensions the $xy$ plane divides space into the regions $z>0$ and $z<0$.

You can see the algebraic analog for any number of coordinates, even though it's hard to "see" above and below geometrically.

Then remember that all the hyperplanes behave the same way, since that behavior does not depend on how you set up a coordinate system.