Question

Give the cylindrical and spherical coordinates for the point with rectangular coordinates$(1,0,0)$

Short answer

Cylindrical coordinates are $\left(1,0,0\right)$and spherical coordinates are$\left(1,0,\frac{\mathrm{\pi }}{2}\right)$

Step-by-step solution

Given Information

The given coordinates in rectangular coordinates are$\left(1,0,0\right)$

Finding the cylindrical coordinates

It is given that$x=1,y=0,z=0$

We know that $r=\sqrt{x^2+y^2},\theta ={\tan }^{-1}\frac{y}{x},z=z$

Substituting the values, we get

$r=\sqrt{1^2+0^2},\theta ={\tan }^{-1}\frac{0}{1},z=0$

$\Rightarrow r=1,\theta =0°,z=0$

localid="1651608209346" $\Rightarrow \left(r,\theta ,z\right)=(1,0,0)$

Finding the spherical coordinates

We know that $\rho =\sqrt{x^2+y^2+z^2},\theta ={\tan }^{-1}\frac{y}{x},\varphi ={\cos }^{-1}\frac{z}{\sqrt{x^2+y^2+z^2}}$

$\Rightarrow \rho =\sqrt{1^2+0^2+0^2},\theta ={\tan }^{-1}\frac{0}{1},\varphi ={\cos }^{-1}\frac{0}{\sqrt{1^2+0^2+0^2}}$

$\Rightarrow \rho =1,\theta =0°,\varphi ={\cos }^{-1}0°$

$\Rightarrow \rho =1,\theta =0°,\varphi =\frac{\mathrm{\pi }}{2}$

$\Rightarrow \left(\rho ,\theta ,\varphi \right)=\left(1,0,\frac{\mathrm{\pi }}{2}\right)$