Q 27. - Calculus

Question

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

Step-by-Step Solution

Answer

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

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

Step 1: Given Information

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

Step 2: 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

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

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

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

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

Step 3: 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

$r=8\mathrm{sin\pi }, \theta =\frac{\mathrm{\pi }}{2}, z=8\mathrm{cos\pi }$

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

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

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