If the volume of the parallelopiped formed by the vectors $\vec a,\vec b,\vec c$ as three coterminous edges is 27 cubic units, then the volume of the parallelopiped having $\vec α = \vec a +2\vec b-\vec c, \vec β=\vec a-\vec b$ and $\vec γ =\vec a-\vec b-\vec c$ as three coterminous edges, is

81 cubic units

We have, $\left|[\vec a\,\,\vec b\,\,\vec c]\right|= 27$ cubic units

Now, $[\vec α\,\,\vec β\,\,\vec γ]=\begin{vmatrix}1&2&-1\\1&-1&0\\1&-1&-1\end{vmatrix}[\vec a\,\,\vec b\,\,\vec c]$

$⇒[\vec α\,\,\vec β\,\,\vec γ]=3[\vec a\,\,\vec b\,\,\vec c]$

∴ Required volume = $\left|[\vec α\,\,\vec β\,\,\vec γ]\right|$

$=3\left|[\vec a\,\,\vec b\,\,\vec c]\right|=3×27=81$ cubic units