If $\vec a,\vec b,\vec c$ are three non-coplanar vectors represented by non-current edges of a parallelopiped of volume 4 units, then the value of $(\vec a+\vec b). (\vec b×\vec c) + (\vec b + \vec c). (\vec c×\vec a) + (\vec c + \vec a). (\vec a×\vec b)$, is

±12

We have, $[\vec a\,\,\vec b\,\,\vec c]=±4$

$∴(\vec a+\vec b). (\vec b×\vec c) + (\vec b + \vec c). (\vec c×\vec a) + (\vec c + \vec a). (\vec a×\vec b)$

$=3[\vec a\,\,\vec b\,\,\vec c]=±12$