Let $\vec a,\vec b,\vec c$ be three non-zero non-coplanar vectors and p, q and r be three vectors given by $\vec p=\vec a+\vec b-2\vec c, \vec q=3\vec a-2\vec b+\vec c$ and $\vec r = \vec a -4\vec b+2\vec c$. If the volume of the parallelopiped determined by $\vec a, \vec b$ and $\vec c$ is $V_1$ and that of the parallelopiped determined by $\vec a, \vec q$ and $\vec r$ is $V_2$, then $V_2: V_1=$

15 : 1

We have, $V_1 =[\vec a\,\,\vec b\,\,\vec c]$ and, $V_2 = [\vec p\,\, \vec q\,\, \vec r]$

Now,

$V_2 = [\vec p\,\, \vec q\,\, \vec r]$

$⇒V_2 =\begin{vmatrix}1&1&-2\\3&-2&1\\1&-4&2\end{vmatrix}[\vec a\,\,\vec b\,\,\vec c]$

$⇒V_2 =15 V_1 ⇒V_2 :V_1 = 15:1$