Vector Orthogonality

Orthogonalization

Finding any Perpendicular Vector

Finding any perpendicular vector to a vector v amounts to finding another vector u, where

u != v

A perpendicular vector to v can then be found:

p = u x v

So, in pseudocode, this could look like:

if v == [1,0,0]
    p = [0,1,0] x v
else
    p = [1,0,0] x v

Perpendicular Vector Close to other Vector

If we have a vector v and another vector u, and we want to find the vector p closest to u, that is perpendicular to v, we can use the simplest case of Gram-Schmidt orthogonalization:

p = u - (u projected onto v)
  = u - (u dot norm(v)) * norm(v)

This can be described as the vector from the projected vector to u, which must be perpendicular to v:

   / u
  /
 /
/
--->---> v
   |
 u projected onto v

   /
  /|   u - (u onto v)
 / | - the closest vector to u that is orthogonal to v
/  |
---*--->