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

Oct 23, 2016

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

#### Explanation:

South latitudes and West longitudes are negatives.

In spherical-polar coordinates, the position vectors to the

locations

$P \left(6371 , {34}^{o} , - {34}^{o}\right) \mathmr{and} Q \left(9371 , - {34}^{o} , {34}^{o}\right)$ 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