Computing two vectors that are perpendicular to third vector in 3D

Question

What is the best (fastest) way to compute two vectors that are perpendicular to the third vector(X) and also perpendicular to each other?

This is ho

Answer 1

What I have done, provided that X<>0 or Y<>0 is

1. A = [-Y, X, 0]
2. B = [-X*Z, -Y*Z, X*X+Y*Y]

and then normalize the vectors.

[ X,Y,Z]·[-Y,X,0] = -X*Y+Y*X = 0
[ X,Y,Z]·[-X*Z,-Y*Z,X*X+Y*Y] = -X*X*Z-Y*Y*Z+Z*(X*X+Y*Y) = 0
[-Y,X,0]·[-X*Z,-Y*Z,X*X+Y*Y] = Y*X*Z+X*Y*Z = 0

This is called the nullspace of your vector.

If X=0 and Y=0 then A=[1,0,0], B=[0,1,0].