Proof of using dot product to determine which side of a vector a point is on For illustration sake. 
Lets set $r=[3,3]$ and $s=[4,8]$
Given points $r$,$s$ and a vector $\underline v$, 
How do I prove that the point $r$ is on one side of the plane?
I know that you can use a dot product to prove this. 
For example, if $r\cdot\underline v > 0$ it is on one side of the vector. If $r\cdot\underline v < 0$, it is on the other side. 
The intuition seems logical to me but I cannot seem to prove it myself with real working numbers and examples. 
Can anyone prove it to me? I have looked at other similar questions but I cannot seem to understand them. 

           


Perhaps what you mean is that $v$ defines a halfplane: $v$ is orthogonal
to the line bounding the halfplane. Let me assume the halfplane line
passes through the origin. Then when $p \cdot v > 0$, $p$ lies in the
positive halfplane. When $p \cdot v < 0$, it lies on the negative side.
And $p \cdot v = 0$ defines the halfplane bounding line.
