Dick Fischbeck wrote:To deal with half-spaces, one pretty much needs the concept of 'orthogonal to a given vector'.
A half-space in our three-dimensional world would be some plane and all of the volume below it. A typical way to represent a plane in three-dimensions is to specify a vector which is perpendicular to all lines in the plane and then specify a point that is in the plane. Orthogonal and perpendicular mean the same thing (at least in Euclidean geometry).
So, for example, I might divide the Universe into two halves. There is all of the Universe that is above my floor and all of the Universe that is below my floor. Then, I could specify this by saying that I have the vector: < 0, 1, 0 > which points directly up... and then I would also need to specify a point on my floor (so we can distinguish that I was really talking about the floor and not the ceiling or some other parallel plane).
In n-dimensional space, any hyperplane divides the space into two halves. We can specify the hyperplane by telling one point in the hyperplane and giving a vector which is perpendicular to all lines in the hyperplane.
More later if this doesn't clear things up...