Here we illustrate the concept of a hyperspace in \(\RNrSpc{2}\) or in \(\RNrSpc{3}\).
A hyperspace in \(\RNrSpc{2}\) is just a line through the origin. Thus the red line in the picture below is a hyperspace. We characterize it as the collection of all those vectors which are perpendicular to the green vector \(\Vect{a}\), called a normal vector to the hyperspace.
Notice that a hyperspace has many normal vectors: if \(\Vect{a}\) is a normal vector, so are \(2\cdot \Vect{a}\), \(3\cdot \Vect{a}\), \((-1)\cdot \Vect{a}\) and, in general, \(t\cdot \Vect{a}\) for any \(t\neq 0\).
A hyperspace in \(\RNrSpc{3}\) is just a plane, in the usual sense of the word, which passes through the origin.
In the picture above, if we call the hyperspace \(H\), then the blue arrow \(\Vect{a}\), is called a normal vector of \(H\) because it is perpendicular to all those vectors which belong to \(H\). The second plane in this picture (horizontal) is only there to provide visual reference points. Note that any non-zero multiple of \(\Vect{a}\) is also a normal vector of \(H\): if \(n\geq 1\), there are always many normal vectors to a given hyperspace.
If \(n \geq 4\), we cannot visualize a hyperspace in \(\RNrSpc{n}\). Still we can characterize it as the collection of all those vectors in \(\RNrSpc{n}\) which are perpendicular to a given nonzero vector \(\Vect{a}\); that is as vectors satisfying the equation \(\DotPrVecs{a}{x}=0\).