Why swear in the vice president first?
The simple explanation: The US Constitution does not permit that
the nation be without a chief executive. Therefore it is a
practical matter, considering that the Vice president can act as
the chief executive when the President is incapasitated. For the
few moments that the former President is stepping down, and the new
President is being sworn in, there would actually be no President,
so to avoid the problem, the Vice President is traditionally sworn
in first.