Arc length and angle between two cities

[lamerali]

Two cities on the surface of the Earth are represented by position vectors that connect the location of each city to the centre of the earth. Assuming that the centre of the Earth is assigned the coordinates of the origin, and that the Earth is a perfect sphere, outline the steps that would lead to a calculation of the shortest distance between the two cities. Hint: How can you determine arc length?


Would I just find the cross product between the radius of Earth and the angle between the two cities since arc length = radius x angle in rads? if so how do i find the angle between the two cities in the first place?

thanks in advance,
Layla
Calculus and vectors

 

[HallsofIvy]

Well, first, you don't mean "cross product" because that is a product between two vectors, not two numbers ("the radius of Earth and the angle between the two cities).
However, yes, if you know the angle between two cities (in radians), multiplying that by the radius of the Earth will give the distance between them. As for finding the angle between two points, given the latitude and longitude of each, that's "spherical trigonometry". I don't have the time to go through it right now but look at this website:
http://mathworld.wolfram.com/SphericalTrigonometry.html

 

[lamerali]

Thanks!