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Worked solution to challenge 1

Let d be the distance in kilometres between London and New York and W be the angle subtended by the two cities at the centre of the Earth
Let r1, q1, f1 be the spherical coordinates of London and r2, q2, f2 be those of New York
Some thought should convince you that latitude = 90 - q, hence from the latitudes and longitudes of the two cities:

r1 = 6369 r2 = 6369
q1 = 38 q2 = 49
f1 = 0 f2 = -73

From earlier we know that we can change from spherical polar to cartesian coordinates using,
x = r sin q cos f
y = r sin q sin f
z = r cos q
hence,

x1 = 3921.15 x2 = 1405.36
y1 = 0 y2 = -4596.71
z1 = 5018.84 z2 = 4178.44

Taking the dot product gives the value,
x1x2 + y1y2 + z1z2 = 26481549.17
But the alternative definition of the dot product is:
p1.p2 = |p1||p2| cos W
where p1 is the position vector of London and p2 the position vector of New York
|p1| = |p2| = r1 = r2 = 6369
Hence, W = 49.2446 degrees
Converting to radians (*p/180), W = .859480 radians
d = W * radius of the earth = .859480 * 6369 = 5474 km