Gps Great Circle Distance

Cant figure out this physics problem?

2. (10 points) An airliner flies from Washington D.C. to Manila along a great circle route.
This route, an arc of a circle which has its center at the center of the earth, is the shortest
distance across the surface of the globe between any two locations. What distance, in
kilometers, is covered by the plane during this flight? The GPS coordinates of the two
cities, to three significant figures, are 38.9º N, 77.0º W and 14.6º N, 121º E, respectively.
Important: You are expected to show, starting from basic principles, how to calculate the
great circle distance between any two points on the globe with known coordinates – travel
computation formulas from handbooks or internet sites are not acceptable. (Hint: It might
help to review spherical coordinates.)

We need the angle subtended between these two locations

First convert each to it spherical coordinates (r, θ, φ) where θ is the longitude and φ is the latitude

So converting to x,y,z we use

y = R*sin(φ), xz = R*cos(φ) so x = R*cos(φ)*cos(θ) and z = R*cos(φ)*cos(θ)

So for x y z for Washington is x = R*cos(38.9)*cos(77) = R*0.1751

y = R*sin(38.9) = R*0.6280

z = R*cos(38.9)*sin(77) = R*0.7583

For Manila x = R*cos(14.6)*cos(-121) = -0.4984*R

y = R*sin(14.6) = R*0.2521

z = R*cos(14.6)*sin(-121) = -0.8295*R

Now the angle between them is the arccos of the dot product of the two vectors divided by the products of their length..Note here L = R

So θ = arccos((0.1751R*(-0.4984R) + 0.6280R*0.2521R + 0.7583R*(-0.8295R)/R^2

So θ = arccos(-0.5580) = 123.9o

So the distance between them along the surface of the Earth is R*123.9*π/180

= 1.38×10^7m = 1.38×10^4 km

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