Q 23. - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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

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

$\Rightarrow r=1, \theta =0°, z=0$

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

Step 3: Finding the spherical coordinates

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

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

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

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

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