If $\overline{a}$ is perpendicular to both $\overline{b}$ and $\overline{c}$ such that $\left|\overline{a}\right|=2,\left|\overline{b}\right|=3,\left|\overline{c}\right|=4$ and the angle between $\overline{b}and\overline{c}\text{ is}\frac{2\pi }{3},$ then $\left|\left[\overline{a}\overline{b}\overline{c}\right]\right|$ is equal to

$\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}=\overline{\mathrm{a}}\cdot \overline{\mathrm{c}}=0,\overline{\mathrm{b}}.\overline{\mathrm{c}}=\left|\overline{\mathrm{b}}\right|\left|\overline{\mathrm{c}}\right|\cos \frac{2\pi }{3}=3\times 4\times \frac{-1}{2}=-6$

So

${\left[\begin{array}{ccc}\overline{\mathrm{a}}& \overline{\mathrm{b}}& \overline{\mathrm{c}}\end{array}\right]}^2=\left|\begin{array}{l}\overline{\mathrm{a}}\cdot \overline{\mathrm{a}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{\mathrm{a}},\overline{\mathrm{b}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{\mathrm{a}}\cdot \overline{\mathrm{c}}\\ \overline{\mathrm{b}}\cdot \overline{\mathrm{a}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{\mathrm{b}}\cdot \overline{\mathrm{b}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{\mathrm{b}}\cdot \overline{\mathrm{c}}\\ \overline{\mathrm{c}}\cdot \overline{\mathrm{a}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{\mathrm{c}}\cdot \overline{\mathrm{b}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\overline{\mathrm{c}}\cdot \overline{\mathrm{c}}\end{array}\right|=\left|\begin{array}{ccc}4& 0& 0\\ 0& 9& -6\\ 0& -6& 16\end{array}\right|$

$=4(144-36)=4\times 108=432$

$=\left[\begin{array}{ccc}\overline{\mathrm{a}}& \overline{\mathrm{b}}& \overline{\mathrm{c}}\end{array}\right]=2\times 2\times 3\sqrt{3}=12\sqrt{3}$