Answer
a) $(2, \dfrac{3\pi}{2},\dfrac{\pi}{2})$
b) $(2, \dfrac{3\pi}{4},\dfrac{3\pi}{4})$
Work Step by Step
Conversion of rectangular to spherical coordinates as follows:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and $\rho=\sqrt {x^2+y^2+z^2}$; $\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$
a) Here, $\rho=\sqrt {0^2+)(-2)^2+0^2}=2$
$\cos \phi =\dfrac{0}{2} \implies \phi=\dfrac{\pi}{2}$; $\cos \theta=\dfrac{0}{2 \sin \dfrac{\pi}{2}} \implies \theta=\dfrac{3\pi}{2}$
Thus, $(r, \theta, \phi)=(2, \dfrac{3\pi}{2},\dfrac{\pi}{2})$
b) Here, $\rho=\sqrt {(-1)^2+)(1)^2+(-\sqrt2)^2}=2$
$\cos \phi =\dfrac{-\sqrt 2}{2} \implies \phi=\dfrac{3 \pi}{4}$; $\cos \theta=\dfrac{-1}{2 \sin \dfrac{3\pi}{4}} \implies \theta=\dfrac{3\pi}{4}$
Thus, $(r, \theta, \phi)=(2, \dfrac{3\pi}{4},\dfrac{3\pi}{4})$
