## Thomas' Calculus 13th Edition

${\bf u}\times{\bf v}$ has length $1$ and direction ${\bf j}$ ${\bf v}\times{\bf u}$ has length $1$ and direction $-{\bf j}$
${\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ u_{1} & u_{2} & u_{3}\\ v_{1} & v_{2} & v_{3} \end{array}\right|$ $=(u_{2}v_{3}-u_{3}v_{2}){\bf i}-(u_{1}v_{3}-u_{3}v_{1}){\bf j}+(u_{1}v_{2}-u_{2}v_{1}){\bf k}$ --- ${\bf u}={\bf i}\times{\bf j}={\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{array}\right|={\bf k}$ ${\bf v}={\bf j}\times{\bf k}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array}\right|={\bf i}$ ${\bf w}={\bf u}\times{\bf v}=\left|\begin{array}{lll} {\bf i} & {\bf j} & {\bf k}\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{array}\right|=-(0-1){\bf j}={\bf j}$ $\left|{\bf w}\right|=1,$ ${\bf v}\times{\bf u}=-{\bf w}=1\cdot(-{\bf j})$ ${\bf u}\times{\bf v}$ has length $1$ and direction ${\bf j}$ ${\bf v}\times{\bf u}$ has length $1$ and direction $-{\bf j}$