What is the ruling on bowing to a scholar when shaking hands with him, and what is the jurists' opinion on that? Is the ruling related to the intention of the one shaking hands?

Bending is not prescribed when shaking hands with a scholar or anyone else, due to the saying of the Messenger of Allah, peace and blessings be upon him, to one who asked him: "A man among us meets his brother or friend, should he bow to him?" He said: "No."

And if bending as a form of greeting takes the form of prostration, it is forbidden, and it is considered polytheism (shirk) if it is intended to glorify a creature as Allah is glorified.