   Chapter 15.8, Problem 46E

Chapter
Section
Textbook Problem

The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates ρ, θ ,ϕ 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 α = 90° - ϕ° and the longitude is β = 360° - θ °. Find the great-circle distance from Los Angeles (lat. 34.06° N, long. 118.25° W) to Montréal (lat. 45.50° N, long. 73.60° W). 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.)

To determine

To find: The great circle distance from Los Angeles to Montreal.

Explanation

Given:

The latitude and longitude of Los Angeles is respectively 34.06°N and 118.25°W .

The latitude and longitude of Montreal is respectively 45.50°N and 73.60°W .

The latitude of the point P is α=90°ϕ° and the longitude of the point P is β=360°θ° .

The radius of the earth is 3960 mi.

Formula used:

The great circle distance between any two coordinate of three dimensional space is given by, d=σ×π×ρ180° . Here, the angle between the given points is given by σ , the radius of the Earth is given by ρ (1)

The dot product of two given points is A.B=|A|×|B|×cos(σ) (2)

The spherical coordinates (ρ,θ,ϕ) corresponding to the rectangular coordinates (x,y,z) is,

ρ=x2+y2+z2ϕ=cos1(zρ)θ=cos1(xρsinϕ)

Calculation:

Let A and B represents the three dimensional coordinates of the Los Angeles and Montreal respectively that is A=(x1,y1,z1),B=(x2,y2,z2) . Let the radius of the earth be denoted by ρ . Convert the coordinates into spherical coordinates by the above mentioned formula. Thus, A=(ρsinϕ1cosθ1,ρsinϕ1sinθ1,ρcosϕ1) and B=(ρsinϕ2cosθ2,ρsinϕ2sinθ2,ρcosϕ2) . ρ is not varying because it denotes the radius of the earth. With the given conditions, find the corresponding latitude and longitude for the point P as shown below.

α=90°ϕ1°34.06°=90°ϕ1°34.06°90°=ϕ1°ϕ1°=55.94°

α=90°ϕ2°45.50°=90°ϕ2°45.50°90°=ϕ2°ϕ2°=44

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