Question

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

Short answer

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

Cylindrical coordinates arelocalid="1651612310902" $\left(r,\theta ,z\right)=\left(0,\frac{\mathrm{\pi }}{2},-8\right)$

Step-by-step solution

Given Information

The given expression is $\left(\rho ,\theta ,\varphi \right)=\left(8,\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$

To find rectangular coordinates

As $\rho =8,\theta =\frac{\mathrm{\pi }}{2},\varphi =\mathrm{\pi }$

We know that for converting spherical to rectangular coordinates,

$x=\rho \sin \varphi \cos \theta ,y=\rho \sin \varphi \sin \theta ,z=\rho \cos \varphi$

Using values given, we get

localid="1652288939533" $x=8\mathrm{sin\pi cos}\frac{\mathrm{\pi }}{2},y=8\mathrm{sin\pi sin}\frac{\mathrm{\pi }}{2},z=8\mathrm{cos\pi }$

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

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

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

To find cylindrical coordinates

To find cylindrical coordinates, we get

$r=\rho \sin \varphi ,\theta =\theta ,z=\rho \cos \varphi$

Substituting spherical coordinates, we get

role="math" localid="1652289128136" $r=8\mathrm{sin\pi },\theta =\frac{\mathrm{\pi }}{2},z=8\mathrm{cos\pi }$

$r=8\left(0\right),\theta =\frac{\mathrm{\pi }}{2},z=8\left(-1\right)$

$\Rightarrow r=0,\theta =\frac{\mathrm{\pi }}{2},z=-8$

Hence, cylindrical coordinates are$\left(r,\theta ,z\right)=\left(0,\frac{\mathrm{\pi }}{2},-8\right)$