What is the distance, measured along a great circle, between locations at #(34^o N, 34^o W) and (34^ S, 34^o E)#?

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Answer 1 · 2016-10-23T02:09:52

#10360# km #= 10.36# Mm, for 4-sd rounding

South latitudes and West longitudes are negatives.

In spherical-polar coordinates, the position vectors to the

locations

#P(6371, 34^o, -34^o) and Q(9371, -34^o, 34^o)# from the center O

of the Earth are

#OP=6371( cos 34° cos(- 34°), cos 34° sin(- 34^o°), sin 34°# and

#OQ=6371( cos(- 34°) cos 34°, cos(- 34°) sin 34°, sin(- 34°)#,

For unit vectors #n_(OP) and n_(OQ), omit the factor 6371 km (mean

radius of the Earth ). in these normal directions

The angle subtended by the great-circle arc PQ at the center is

#alpha=arc cos(n_(OP)·n_(OQ))#

#=arc cos (cos^4 34°-cos^2 34° sin^2 34°-sin^2 34°)#

#=arc cos(cos^2 34° cos 68°-sin^2 34°)#

#=93.16°#

The great-circle arc distance PQ = 6371 X alpha in radians#

#=6371(93.16/180)\pi)# km

#=10360# km #= 10.36# Mm, for 4-sd rounding