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 r_{1}, q_{1}, f_{1} be the spherical coordinates of
London and r_{2}, q_{2}, f_{2} be those of New York

Some thought should convince you that latitude = 90 - q, hence from the
latitudes and longitudes of the two cities:

r_{1} = 6369 |
r_{2} = 6369 |

q_{1} = 38 |
q_{2} = 49 |

f_{1} = 0 |
f_{2} = -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,

x_{1} = 3921.15 |
x_{2} = 1405.36 |

y_{1} = 0 |
y_{2} = -4596.71 |

z_{1} = 5018.84 |
z_{2} = 4178.44 |

Taking the dot product gives the value,

x_{1}x_{2} + y_{1}y_{2} + z_{1}z_{2} = 26481549.17

But the alternative definition of the dot product is:

__p___{1}.__p___{2} =
|__p___{1}||__p___{2}| cos W

where __p___{1} is the position vector of London and __p___{2}
the position vector of New York

|__p___{1}| = |__p___{2}| = r_{1} = r_{2} = 6369

Hence, W = 49.2446 degrees

Converting to radians (*p/180),
W = .859480 radians

d = W * radius of the earth = .859480 * 6369 = 5474 km