$\text{ The value of }[\overrightarrow{\text{A}}-\overrightarrow{\text{B}},\overrightarrow{\text{B}}-\overrightarrow{\text{C}},\overrightarrow{\text{C}}-\overrightarrow{\text{A}}]\text{ where }\left|\overrightarrow{\text{A}}\right|=1,\left|\overrightarrow{\text{B}}\right|=2\text{ and }\left|\overrightarrow{C}\right|=3\text{ is }$

$$\begin{gathered} \begin{array}{c}\left[\begin{array}{lll}\overrightarrow{\text{A}}-\overrightarrow{\text{B}}& \overrightarrow{\text{B}}-\overrightarrow{\text{C}}& \overrightarrow{\text{C}}-\overrightarrow{\text{A}}]\end{array}\right.=\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right|\left[\begin{array}{ccc}\overline{A}& \overline{B}& \overline{C}\end{array}\right]\\ =0\left[\begin{array}{ccc}A& B& C\end{array}\right]\end{array} \\ =0 \end{gathered}$$