Answer
(a) $ r(u,v)=\lt u \cos v, u \sin v, 1-u \cos v-u \sin v\gt $ and $0 \le u \le 3$; $0 \le v \le 2\pi $
(b) $ r(u,v)=\lt 1-u \cos v-u \sin v, u \cos v, u \sin v\gt $ where $0 \le u \le 3$; $0 \le v \le 2\pi $
Work Step by Step
(a) Apply polar coordinates in the $ xz $ plane.
We know that $ r(r, \theta)=xi+yj+zk $ or, $ r^2=x^2+y^2+z^2$
We have $ x^2+z^2=4$ and $ z=1-(x+y)$
Thus, $ x=r \cos \theta, y=r \sin \theta, z=z $
Hence: $ r(u,v)=\lt u \cos v, u \sin v, 1-u \cos v-u \sin v\gt $ where $0 \le u \le 3$; $0 \le v \le 2\pi $
(b) Apply polar coordinates in the $ xz $ plane.
We know that $ r(r, \theta)=xi+yj+zk $ or, $ r^2=x^2+y^2+z^2$
We have $ y^2+z^2=9$ and $ x=1-(y+z)$
Thus, $ x=r \cos \theta, y=r \sin \theta, z=z $
Hence: $ r(u,v)=\lt 1-u \cos v-u \sin v, u \cos v, u \sin v\gt $ where $0 \le u \le 3$; $0 \le v \le 2\pi $
