The distance between point P1(ρ1, θ1, ϕ1) and P2(ρ2, θ2, ϕ2) given in spherical coordinates extract the formula.

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The distance between point P1(ρ1, θ1, ϕ1) and P2(ρ2, θ2, ϕ2) given in spherical coordinates extract the formula.

Cheryl King · Accepted Answer

Step 1 Given point P1(ρ1, θ1, ϕ1), P2(ρ2, θ2, ϕ2) Let corresponding coutesian cordinates are, P1(x1, y1, z1) &amp;amp; P2(x2, y2, z2) Relation between coutesian cordinates and sherical coordinates are x1=ρ1, sin⁡θ1, cos⁡ϕ1 y1=ρ1, sin⁡ϕ1, sin⁡θ1 z1=ρ1, cos⁡θ1 x2=ρ2, sin⁡θ2, cos⁡ϕ2 y2=ρ2sin⁡θ2 sin⁡ϕ2 z2=ρ2 cos⁡θ2 Distance between paints P1 &amp;amp; P2 d=(x1−x2)2+(y1−y2)2+(z1−z2)2 =x12+x2−2x1x2+y12+y22−2y1y2+z12+z22−2z1z2 =(x12+y12+z12)+(x22+y22+z22)−2(x1x2+y1y2+z1z

ambarakaq8 · Accepted Answer

Assuming that θ is the latitude and ϕ is the longitude we have that the cartesian coordinates of the first point are:  (ρ1cos⁡θ1cos⁡ϕ1, ρ1cos⁡θ1sin⁡ϕ1, ρ1sin⁡θ1)  so the distance between the two points is given by:  ρ12+ρ22−2ρ1ρ2(cos⁡θ1cos⁡θ2cos⁡(ϕ1−ϕ2)+sin⁡θ1sin⁡θ2}

nick1337 · Accepted Answer

Step 1The expression of the distance between two vectors in spherical coordinates provided in the other response is usually expressed in a more compact form that is not only easier to remember but is also ideal for capitalizing on certain symmetries when solving problems.||r−r′||=(x−x′)2+(y−y′)2+(z−z′)2=r2+r′2−2rr′[sin⁡(θ)sin⁡(θ′)cos⁡(ϕ)cos⁡(ϕ′)+sin⁡(θ)sin⁡(θ′)sin⁡(ϕ)sin⁡(ϕ′)+cos⁡(θ)cos⁡(θ′)]=r2+r′2−2rr′[sin⁡(θ)sin⁡(θ′)(cos⁡(ϕ)cos⁡(ϕ′)+sin⁡(ϕ)sin⁡(ϕ′))+cos⁡(θ)cos⁡(θ′)]=r2+r′2−2rr′[sin⁡(θ)sin⁡(θ′)cos⁡(ϕ−ϕ′)+cos⁡(θ)cos⁡(θ′)]This form makes it fairly transparent how azimuthal symmetry allows you to automatically eliminate some of the angular dependencies in certain integration problemsAnother advantage of this form is that you now have at least two variables, namely ϕ and ϕ′,  that appear in the equation only once, which can make finding series expansions w.r.t. these variables a little less of a pain than the others.

The distance between point P_{1}(\rho_{1},\ \theta_{1},\ \phi_{1}) and P_{2}(\rho_{2},\ \theta_{2},\

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