If a person lives in another state but has committed a crime in Virginia while he was visiting there, will Virginia ask for extradition?

There is no clear cut answer to that. Whether the State of Virginia is going to ask another state to extradite a person depends on two main factors:
1) How severe the crimeis. If it is a minor offense, chances are slim that Virginia authorities will go over the red tape involved in an interstate extradition process.
2) Whether the DA in Virginia achieve conviction in a trial. If the chances for conviction are low, so are the chances for an extradition request.