I want to ask about a simple gyroscope experiment: the one in which a man is holding a gyroscope, and sits on a rotating chair, and when he rotates the gyroscope, he himself starts rotating in regard of the conservation of angular momentum.
For reference, I am putting up this link to Feynman. Initially, only the gyroscope is rotating, and the total angular momentum of the system is a vector pointing in the right direction, which basically is the angular momentum of the gyroscope. Now, the man rotates the gyroscope in such a way that now it is facing in the upwards direction, towards the sky. Here, the gyroscope has an angular momentum in the upwards direction, so to negate this, the man and chair start rotating at the same rate, but in the opposite direction. However, in this case, the angular momentum becomes $0$. It should instead be a vector again pointing in the right direction.
What is the problem that I am making here?