If you are actually using a rotation to find the new facing then you run the up vector through the same rotation. The answer given was more for when you arbitrarily assign a new facing. Lookat which was mentioned is a prime example. You don''t rotate to that direction but rather just switch to it. If your Lookat is actually rotating the camera smoothly to point in that direction then just rotate the up vector along with it. It will still be an orthogonal vector as long as it is a linear transform, i.e. uniform scaling with no shear.