## Distance from Palo Alto to Beijing. The surface of the earth is reasonably approximated as a sphere with radius R = 6367.5km. A location on

Question

Distance from Palo Alto to Beijing. The surface of the earth is reasonably approximated as a sphere with radius R = 6367.5km. A location on the earth’s surface is traditionally given by its latitude θ and its longitude λ, which correspond to angular distance from the equator and prime meridian, respectively. The 3-D coordinates of the location are given by

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2022-01-08T13:42:27+00:00
2022-01-08T13:42:27+00:00 1 Answer
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## Answers ( )

Answer:The 3-D coordinates of the location of a point on the earth’s surface is given by (R₀ₐ ×cosλ ×cosθ, R₀ₐ×sinλ×cosθ, R₀ₐ×sinλ) where λ =longitude and θ = latitude

and the distance between points on the surface such as two cities is given by P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) = |P₁ P₂| = √((x₂-x₁)² + (y₂-y₁)² + (z₂ – z₁)²)

Step-by-step explanation:If the center of the earth is marked O the location of a point is given in longitude and latitude as λ and θ respectively then where normally we have

-180 ° ≤ λ ≤ 180 ° and 90 ° ≤ θ ≤ 90°

However in cartesan coordinates, considering the radius of the earth and specifying the location of a point on the surface of the earth in three dimensions of the x, y and z coordinates with distance from the earth center = R where R = |Oa| = 6367.5km and the center of the Earth is O, then a point, a on the surface of the earth has the x, y and z Cartesian coordinates we have

a = (xₐ, yₐ, zₐ) = (R×cosλ×cosθ, R×sinλ×cosθ, R×sinλ)

Hence the 3-D coordinates of a location is given by

(R₀ₐ ×cosλ ×cosθ, R₀ₐ×sinλ×cosθ, R₀ₐ×sinλ) with R₀ₐ = the distance of the point a to the center of the earth

Distance between two points is given by P₁ = (x₁, y₁, z₁) and P₂ = (x₂, y₂, z₂) = |P₁ P₂| = √((x₂-x₁)² + (y₂-y₁)² + (z₂ – z₁)²)