Distance of a Point from a Hyperplane

Here we learn how to determine the distance of a point from a hyperplane. This is a very practical problem which one can encounter in many different guises. Later, we will use in a different way within the context of least squares approximations. Happily, there is one simple formula which answers this problem completely and in all dimensions.

Proposition   Let    be the hyperplane of all  x  in    with

(x p) n 0,     p, n  in  ,   n   0.

Then the distance of a point  , with position vector  q, from    is

dist(,)    |(q – p) n|
|n|

The point    on    which is closest to  , has position vector

r   q –  (q – p) n  · n    
n n

Here is a sketch of the situation we are considering:

Exercises   on finding the distance of a point from a hyperplane.

  Move on to any one of the following applications of the dot product