In coordinate space, let $\vec { a } , \vec { b } , \vec { c }$ be the position vectors of three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ with respect to the origin. The dot products between these vectors are shown in the following table.
\begin{center}
\begin{tabular}{ | c | | c | c | c | }
\hline
$\cdot$ & $\vec { a }$ & $\vec { b }$ & $\vec { c }$ \\
\hline
$\vec { a }$ & 2 & 1 & $- \sqrt { 2 }$ \\
\hline
$\vec { b }$ & 1 & 2 & 0 \\
\hline
$\vec { c }$ & $- \sqrt { 2 }$ & 0 & 2 \\
\hline
\end{tabular}
\end{center}
For example, $\vec { a } \cdot \vec { c } = - \sqrt { 2 }$. Which of the following correctly shows the order of the distances between the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$? [4 points]\\
(1) $\overline { \mathrm { AB } } < \overline { \mathrm { AC } } < \overline { \mathrm { BC } }$\\
(2) $\overline { \mathrm { AB } } < \overline { \mathrm { BC } } < \overline { \mathrm { AC } }$\\
(3) $\overline { \mathrm { AC } } < \overline { \mathrm { AB } } < \overline { \mathrm { BC } }$\\
(4) $\overline { \mathrm { BC } } < \overline { \mathrm { AB } } < \overline { \mathrm { AC } }$\\
(5) $\overline { \mathrm { BC } } < \overline { \mathrm { AC } } < \overline { \mathrm { AB } }$