Lets say we have two sets, $X,Y$, that are separated (in the strict sense) by some hyperplane. (I'll talk about in 3-dimensions as that's how I am trying to picture it, but you can talk about more if you'd like)
I believe then, that there is some vector, $A$, orthogonal to the separating hyperplane, such that the dot product of $A$ with any $x\in X$ is less than the dot product of $A$ with any $y\in Y$ (I believe this is true but am not sure; If this is not true, then instead consider two hyperplanes that satisfy the criteria for the hyperplane separation theorem, and thus there is such a vector, and let me know this isn't true please)
Then, when it is stated that $\langle A,x\rangle < \langle A,y\rangle $ for all $x \in X$ and $y \in Y$, is this saying that:
If I consider the amount that $A$ moves in the direction of $x$, multiplied by the length of $x$, this is less than the amount that $A$ moves in the direction of $y$, multiplied by the length of $y$, $\forall x\in X\; y\in Y$?
(more formally: The magnitude of $A$ projected onto $x$ (times the magnitude of $x$) is less than the magnitude of $A$ projected onto $y$ (times the magnitude of $y$) $\forall x\in X\; y\in Y$?)
Also, when I consider these lengths, do I take the origin to be on the hyperplane? I ask because often when I see hyperplanes depicted I see $A$, the vector orthogonal to the plane, drawn starting at the hyperplane. I realize this may be for easily depicting that it is orthogonal, but I am not sure whether this is the case or if this indicates moving the origin to the hyperplane.
I realize what I take as the origin may not matter for computation purposes, but I asked for purposes of visualization.