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