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

Answered

2021-12-28

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

Answer & Explanation

karton

karton

Expert

2022-01-02Added 439 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

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?