Math for simple 3D coordinate rotation (python)

note: A nicer looking and correct answer will still get accepted, thanks!

I've read on page 27 here that a 3x3 transform matrix can be just the nine dot products - thank you U. Auckland's prof. Kelly!

above x2: screenshots from here.

Here is a very ugly implementation which seems to work.

new_yaxis = -np.cross(new_xaxis, new_zaxis)

# new axes:
nnx, nny, nnz = new_xaxis, new_yaxis, new_zaxis
# old axes:
nox, noy, noz = np.array([1, 0, 0, 0, 1, 0, 0, 0, 1], dtype=float).reshape(3, -1)

# ulgiest rotation matrix you can imagine
top = [np.dot(nnx, n) for n in [nox, noy, noz]]
mid = [np.dot(nny, n) for n in [nox, noy, noz]]
bot = [np.dot(nnz, n) for n in [nox, noy, noz]]

def newit(vec):
    xn = sum([p*q for p,q in zip(top, vec)])
    yn = sum([p*q for p,q in zip(mid, vec)])
    zn = sum([p*q for p,q in zip(bot, vec)])
    return np.hstack((xn, yn, zn))

Let's see what happens...

nnx:         array([-0.22139284, -0.73049229,  0.64603887])
newit(nnx):  array([ 1.,  0.,  0.])

nny:         array([ 0.88747002,  0.1236673 ,  0.44396325])
newit(nny):  array([ 0.,  1.,  0.])

nnz:         array([-0.40420561,  0.67163042,  0.62091095])
newit(nnz:   array([ 0.,  0.,  1.])

OK then, this seems to be the right way to go.