3D math challenge

i am trying to figure out:

if i have a XYZ vector, how can i find out for which plane this vector is Z vector. in other words, what are the X and Y vectors for the plane my vector is Z vector.

the answer needed is in vvvvv form ofc (of course).

i am working on it but feel kinda dazed right now, so this is a race.


i am sure, you had that in school… if you have the vector n and you are looking for the vectors a and b, pairwise perpendicular, you have to solve the following equations:

nxax + nyay + nzaz = 0
bx + nyby + nzbz = 0
axbx + ayby + az*bz = 0

theoretically, but do you have a point on the plane?

thatsit! tnx.
you know, whenever i try to remember my skool math, my teacher, and she was really 3D, comes in the way, and she is quite hard to avoid…
darn, m8, gotta concentrate…