## Thomas' Calculus 13th Edition

${\bf u}\times{\bf v}$ has length $0$ and no direction ${\bf v}\times{\bf u}$ has length $0$ and no direction
${\bf u}\times{\bf v}={\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ u_{1} & u_{2} & u_{3}\\ v_{1} & v_{2} & v_{3} \end{array}\right|$ $=(u_{2}v_{3}-u_{3}v_{2}){\bf i}-(u_{1}v_{3}-u_{3}v_{1}){\bf j}+(u_{1}v_{2}-u_{2}v_{1}){\bf k}$ --- ${\bf w}={\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 2 & -2 & 4\\ -1 & 1 & -2 \end{array}\right|$ $=(4-4){\bf i}-(-4+4){\bf j}+(2-2){\bf k}$ $={\bf 0}$ (zero vector has length 0 and no direction) ${\bf v}\times{\bf u}=-{\bf w}=(-{\bf 0})={\bf 0}$ ${\bf u}\times{\bf v}$ has length $0$ and no direction ${\bf v}\times{\bf u}$ has length $0$ and no direction