Answered: "Change from rectangular to spherical coordinates. (0, -2,..."

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2021-11-10

Change from rectangular to spherical coordinates. (0, -2, 0)

Answer & Explanation

Sculd1987

Beginner2021-11-11Added 19 answers

The relationship between rectangular coordinates (x,y,z) and spherical coordinates
$x = ρ \sin (ϕ) \cos (θ)$
$y = ρ \sin (ϕ) \sin (θ)$
$z = ρ \cos (ϕ)$
Let $(x, y, z) = (0, - 2, 0)$
$ρ = \sqrt{x^{2} + y^{2} + z^{2}}$
From above, we can solve for $θ$
$z = ρ \cos (ϕ)$
$\cos (ϕ) = \frac{z}{p} = \frac{0}{2} = 0 \to ϕ = \frac{π}{2}$
Now that we have $ρ$ and $ϕ$ , we can solve for $θ$ $θ$ is not $\frac{π}{2}$ because $y = - 2 < 0$
$\cos (θ) = \frac{x}{ρ \sin (ϕ)} = \frac{0}{2 \sin (\frac{π}{2})} = 0 \to θ = \frac{3 π}{2}$
Result: $(2, \frac{3 π}{2}, \frac{π}{2})$

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