Chapter 13: Q 27. (page 1066)

Give the rectangular and cylindrical coordinates for the point with spherical coordinates $(8, \frac{π}{2}, π)$

Short Answer

Expert verified

Rectangular coordinates are $(x, y, z) = (0, 0, - 8)$

Cylindrical coordinates arelocalid="1651612310902" $(r, θ, z) = (0, \frac{π}{2}, - 8)$

Step by step solution

Given Information

The given expression is $(ρ, θ, ϕ) = (8, \frac{π}{2}, π)$

To find rectangular coordinates

As $ρ = 8, θ = \frac{π}{2}, ϕ = π$

We know that for converting spherical to rectangular coordinates,

$x = ρ \sin ϕ \cos θ, y = ρ \sin ϕ \sin θ, z = ρ \cos ϕ$

Using values given, we get

localid="1652288939533" $x = 8 sinπcos \frac{π}{2}, y = 8 sinπsin \frac{π}{2}, z = 8 cosπ$

$x = 8 (0) (0), y = 8 (0) (1), z = 8 (- 1)$

$x = 0, y = 0, z = - 8$

Hence, required coordinate are $(x, y, z) = (0, 0, - 8)$

To find cylindrical coordinates

To find cylindrical coordinates, we get

$r = ρ \sin ϕ, θ = θ, z = ρ \cos ϕ$

Substituting spherical coordinates, we get

role="math" localid="1652289128136" $r = 8 sinπ, θ = \frac{π}{2}, z = 8 cosπ$

$r = 8 (0), θ = \frac{π}{2}, z = 8 (- 1)$

$\Rightarrow r = 0, θ = \frac{π}{2}, z = - 8$

Hence, cylindrical coordinates are $(r, θ, z) = (0, \frac{π}{2}, - 8)$

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Give the rectangular and cylindrical coordinates for the point with spherical coordinates $(8, \frac{π}{2}, π)$

Short Answer

Step by step solution

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To find rectangular coordinates

To find cylindrical coordinates

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Step by step solution

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Give the rectangular and cylindrical coordinates for the point with spherical coordinates $(8, \frac{π}{2}, π)$