The latitude and longitude of a point P in the Northern Hemisphere are related t

Question

The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates ; ; as follows. We take the origin to be the center of the Earth and the positive z-axis to pass through the North Pole. The positive x-axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of P is = 90 and the longitude is = 360 . Find the great circle distance from Town A (lat. 34.06N, long. 118.25W) to Town B (lat. 37.71N, long.118.25W). Take the radius of the earth to be 3960 mi. (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)

Explanation / Answer

Solution:

The -coordinate of Los Angeles is = (3960; 360 -118:25; 90-34:06) = (3960; 241.75; 55.94).

Thus, the x; y; z-coordinate of Los Angeles is (x; y; z) =

=(3960 sin (55.94) cos (241.75); 3960 sin (55.94) sin (241.75); 3960 cos (55.94)) = (2237.9; 344.26; 3248.8).

Similarly, the ; ; -coordinate of MontrÈal is (; ; ) = (3960; 360 -73.60; 90- 45.50) = (3960; 286.4; 44.5).

Thus, the x; y; z-coordinate of MontrÈal is (x; y; z) =

(3960 sin (44.5) cos (286.4),3960 sin (44.5) sin (286.4), 3960 cos (44:5)) = (-1705,-965.36; 3441.1).

Let ~v be the vector from the origin (center of the earth) to Los Angeles and ~w be the vector from the origin (center of the earth) to MontrÈal. By using the dot product, we have ~v ~w = j~vj j ~wj cos ,

where is the angle between these two vectors which lie on a great circle.

So, = cos-1 [(2237.9; 344.26; 3248.8) (-1705.5; 965.36; 3441.1) ]/(2237.9,-344:26, 3248:8) \ (-1705.5,-965.36;3441.1)

= cos-1 7695000 /3960 x2960 = 0.85465.

Since 360 = 0.85465 2 , we know that is also 48:968 . Therefore, the great circle distance D is 48:968 360 = D 2 (3960), that is, D = 48:968 360 2 (3960) = 3384:4.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.