1. Using spherical coordinates, find the angle
between two arbitrary vectors r? and r?.
Express your answer in terms of ??, ?? and
the difference ?? ? ??. This is the law of
cosines in spherical coordinates. Use your
result to calculate the distance on the Earth's
surface between the Columbia campus and
a) the South Pole.
b) the equator
c) Sydney, Australia.
(Note that latitude and longitude are related
to, but not identical to spherical coordinates.)
Take the radius of the Earth to be 6,371 km.
Show your work so it's clear you did more
than consult ChatGPT. You can use Google or
ChatGPT to find out the longitude and
latitude of the Morningside Heights campus.