Transform the point P (-2, 6, 3) in spherical coordinate system.

Answered question

2021-12-28

Transform the point P (-2, 6, 3) in spherical coordinate system.

Answer & Explanation

karton

karton

Expert2022-01-02Added 613 answers

At point P: x = -2, y = 6, z = 3. Hence

ρ=x2+y2=4+36=6.32

ϕ=tan1yx=tan162=108.43

r=x2+y2+z2=4+36+9=7

θ=tan1=x2+y2z=tan1403=64.42

Thus

P(2,6,3)=P(6.32,108.43,3)=P(7,64.62,108.43)

[ArAθAϕ]=[sinθcosϕsinθsinϕcosϕcosθcosϕcosθsinϕsinθsinθcosθ0][yx+z0]

or

Ar=ysinθcosϕ+(x+z)sinθsinϕ

Aθ=ycosθcosϕ+(x+z)cosθsinϕ

Aϕ=ysinϕ+(x+z)cosϕ

But x=rsinθcosϕ,y=rsinθsinϕ, and z=rcosθ . Substituting these yields

A=(Ar,Aθ,Aϕ)

=r[sin2θcosϕsinϕ+(sinθcosϕ+cosθ)sinθsinϕ]ar+r[sinθcosθsinϕcosϕ+(sinθcosϕ+cosθ)cosθsinϕ]aθ+r[sinθsin2ϕ+(sinθcosϕ+cosθ)cosϕ]aϕ

At P

r=7,tanϕ=62,tanθ=403

Hence, cosϕ=240,sinϕ=640,cosθ=37,sinθ=407

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