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

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Answer 1 · 2016-10-23T02:09:52

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

South latitudes and West longitudes are negatives.

In spherical-polar coordinates, the position vectors to the

locations

$P (6371, 34^{o}, - 34^{o}) and Q (9371, - 34^{o}, 34^{o})$ $P(6371, 34^o, -34^o) and Q(9371, -34^o, 34^o)$ 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

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