Hyperplane - Illustration

Here you can learn how to visualize a hyperplane in    or  .

   A hyperplane in    is a line

The blue line is the hyperspace which is perpendicular to the green normal vector. The red line    is a hyperplane. It results from parallel translating the blue hyperspace.

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   A hyperplane in    is just a plane in the usual sense.

These planes are hyperplanes in 3-space. The origin is supposed to be the footpoint of the two vectors. So the lower hyperplane is even a hyperspace. In general, we characterize the location of a hyperplane by specifying a vector  n  (blue) perpendicular to it and by specifying a point on it (the tip of the red arrow). Alternatively, we can think of a hyperplane as being obtained by parallel translating the hyperspace which is perpendicular to  n  off of the origin by a suitable vector (red).

   In particular, we see that a hyperspace is a very special example of a hyperplane, namely a hyperplane which happens to pass through the origin.

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Return to the main document on hyperplanes.