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