Question

convert the point from rectangular coordinates to spherical coordinates. (-1, 2, 1) (ρ, θ, φ) = ( )

Explanation:

Step1: Calculate $

ho$
The formula for $
ho$ in spherical - coordinates is $
ho=\sqrt{x^{2}+y^{2}+z^{2}}$. Given $x = - 1$, $y = 2$, $z = 1$, then $
ho=\sqrt{(-1)^{2}+2^{2}+1^{2}}=\sqrt{1 + 4+1}=\sqrt{6}$.

Step2: Calculate $\theta$

The formula for $\theta$ is $\theta=\arctan(\frac{y}{x})$. Since $x=-1$ and $y = 2$, $\theta=\arctan(-2)$. But since $x<0$, we add $\pi$ to get $\theta=\pi+\arctan(-2)=\pi-\arctan(2)$.

Step3: Calculate $\varphi$

The formula for $\varphi$ is $\varphi=\arccos(\frac{z}{
ho})$. We know $
ho=\sqrt{6}$ and $z = 1$, so $\varphi=\arccos(\frac{1}{\sqrt{6}})$.

Answer:

$(\sqrt{6},\pi - \arctan(2),\arccos(\frac{1}{\sqrt{6}}))$