The latitude of any location on earth is the angle formed by the two rays drawn from the center of earth to the location and to the equator. the ray through the location is the initial ray. use 3960 miles as the radius of the earth. suppose city a is due north of city b. find the distance between city a (north latitude 45 degrees 28 prime upper n45°28′ n) and city b (latitude 38 degrees 52 prime upper n38°52′ n).

The distance between city a and city b is 833.345 miles.

We know that

1°=60'

The distance of city a from the initial ray is  calculated as

x_a=3960*tan45.46°=4024.101 miles

The distance of city b from the initial ray is calculated as

x_b=3960*tan 38.86°=3190.75 miles

Now the distance between city a and b is equal to

4024.101-3190.75=833.345 miles

This is the vertical distance between the cities.

no. dot product   =0 when vecs at right angles