Answer
$-60$.
Work Step by Step
Put $u=-2i+3j,v=6i-j$, $w=-3i$ in the expression ${{\left\| u+v \right\|}^{2}}-{{\left\| u-v \right\|}^{2}}$. Then,
$\begin{align}
& {{\left\| u+v \right\|}^{2}}-{{\left\| u-v \right\|}^{2}}={{\left\| -2i+3j+6i-j \right\|}^{2}}-\left\| -2i+3j-\left( 6i-j \right) \right\| \\
& ={{\left\| 4i+2j \right\|}^{2}}-{{\left\| -8i+4j \right\|}^{2}}
\end{align}$
Therefore
$\begin{align}
& {{\left\| u+v \right\|}^{2}}-{{\left\| u-v \right\|}^{2}}={{4}^{2}}+{{2}^{2}}-\left( {{\left( -8 \right)}^{2}}+{{4}^{2}} \right) \\
& =16+4-64-16 \\
& =20-80 \\
& =-60
\end{align}$
