## University Calculus: Early Transcendentals (3rd Edition)

$a.$ ${\bf u}\perp{\bf v}$, ${\bf u}\perp {\bf w},$ ${\bf v}\perp{\bf w},$ ${\bf v}\perp{\bf r},$ ${\bf w}\perp{\bf r}$ $b.$ ${\bf u}$ and ${\bf r}$ are parallel.
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.