If \(\bar{a}\) is perpendicular to \(\bar{b}\) and \(\bar{c},|\vec{a}|=2\), \(|\bar{b}|=3,|\bar{c}|=4\) and the angle between \(\bar{b}\) and \(\bar{c}\) is \(\frac{\pi}{3}\), then \(\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]=\)

Let \(\hat{\mathrm{n}}\) be the unit vector perpendicular to \(\overline{\mathrm{b}}\) and \(\bar{c}\).
\({\left[\begin{array}{lll} \overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}} \end{array}\right] } =\overline{\mathrm{a}} \cdot(\overline{\mathrm{~b}} \times \overline{\mathrm{c}}) \)
\( =\overline{\mathrm{a}} \cdot(|\overline{\mathrm{~b}}||\bar{c}| \sin \theta \hat{\mathrm{n}}) \)
\( =\overline{\mathrm{a}} \cdot\left(3 \times 4 \sin \frac{\pi}{3} \cdot \hat{\mathrm{n}}\right) \)
\( =\overline{\mathrm{a}} \cdot\left(12 \times \frac{\sqrt{3}}{2} \hat{\mathrm{n}}\right) \)
\( =6 \sqrt{3}|\overrightarrow{\mathrm{a}}| \hat{\mathrm{n}} \mid \cos 0 \)
\( =6 \sqrt{3} \times 2 \times 1 \)
\( =12 \sqrt{3}\)