Proof for “the hyperplane         is orthoronal to the vector .”
A hyperplane is defined as        .
This  can be also represented as           where  is any point in
 . That is    .
Then, using the orthogonal component of , i.e., ⊥    , we can
express
          ⊥ . Therefore, the hyperplane  consists of an
offset  , plus all vectors orthogonal to the vector .
Now, we choose   ∊  with  ≠ . Then,        .
Thus,    ∊ ⊥ .
Now we claim that    is in the direction of the hyperplane  .
Proof:
Now we prove that    is parallel to the hyperlane or    is in the
direction of the hyperplane. We choose a point  ∊  , and if    is parallel to
the hyperplane  , then a new point         for  ∊  must be also a
point the hyperplane  :
Observe                           , since
     because  ∊  and        as already proved.
Conclusion: Finally, since    is parallel to  and also orthogonal to , the
hyperplane         is orthogonal to the vector .