Answer
$a.$
${\bf u}\perp{\bf v}$,
${\bf u}\perp {\bf w},$
${\bf v}\perp{\bf w},$
${\bf v}\perp{\bf r},$
${\bf w}\perp{\bf r}$
$b.$
${\bf u}$ and ${\bf r}$ are parallel.
Work Step by Step
Two vectors are perpendicular if their dot product is zero.
Two vectors are parallel if their cross product is the zero vector.
$(a)$
${\bf u}\cdot{\bf v}=1(-1)+(2)(1)+(-1)(1)=0$
${\bf u}\cdot{\bf w}=1(1)+(2)(0)+(-1)(1)=0$
${\bf u}\displaystyle \cdot{\bf r}=1(-\frac{\pi}{2})+(2)(-\pi)+(-1)(\frac{\pi}{2})=-3\pi\neq 0$
${\bf v}\cdot{\bf w}=-1(1)+(1)(0)+(1)(1)=0$
${\bf v}\displaystyle \cdot{\bf r}=-1(-\frac{\pi}{2})+(1)(-\pi)+(1)(\frac{\pi}{2})=0$
${\bf w}\displaystyle \cdot{\bf r}=1(-\frac{\pi}{2})+(0)(-\pi)+(1)(\frac{\pi}{2})=0$
${\bf u}\perp{\bf v}$,
${\bf u}\perp {\bf w},$
${\bf v}\perp{\bf w},$
${\bf v}\perp{\bf r},$
${\bf w}\perp{\bf r}$
$(b)$
The perpendicular pairs can not be parallel to each other. We test
${\bf u}\times{\bf r}=\left|\begin{array}{lll}
{\bf i} & {\bf j} & {\bf k}\\
1 & 2 & -1\\
-\pi/2 & -\pi & \pi/2
\end{array}\right|$
$= (\displaystyle \pi-\pi){\bf i}- (\frac{\pi}{2}-\frac{\pi}{2}){\bf j}+(-\pi+\pi){\bf k}={\bf 0}$
${\bf u}$ and ${\bf r}$ are parallel.
