If $\vec a,\vec b$ and $\vec c$ are unit coplanar vectors, then $\begin{bmatrix}2\vec a-3\vec b& 7\vec b-9\vec c& 12\vec c-23\vec a\end{bmatrix}$

0

$\begin{bmatrix}2\vec a-3\vec b& 7\vec b-9\vec c& 12\vec c-23\vec a\end{bmatrix}$

$P=\begin{vmatrix}2&-3&0\\0&7&-9\\-23&0&12\end{vmatrix}[\vec a\,\,\vec b\,\,\vec c]$

as $\vec a\,\,\vec b\,\,\vec c$ are coplanar.

so $[\vec a\,\,\vec b\,\,\vec c]=0$

Hence P = 0 ⇒ given vectors are coplanar as well.