Statement-1: If $G_1, G_2, G_3$ are the centroids of the triangular faces OBC, OCA, OAB of a tetrahedron OABC, then the ratio of the volume of the tetrahedron to that of the parallelopiped with $OG_1, OG_2, OG_3$ as coterminous edges is 9:4.

Statement-2: For any three vectors $\vec a, \vec b, \vec c$

$\begin{bmatrix}\vec a+\vec b&\vec b+\vec c&\vec c+\vec a\end{bmatrix}=2[\vec a\,\,\vec b\,\,\vec c]$

Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Clearly, statement-2 is true.

With reference to the origin O, let the position vectors of the vertices A, B, C be $\vec a, \vec b$ and $\vec c$ respectively. Then, the position vectors of $G_1, G_2, G_3$ are $\frac{\vec b+\vec c}{3},\frac{\vec c+\vec a}{3},\frac{\vec a+\vec b}{3}$ respectively.

Let $V_1$ be the volume of the tetrahedron OABC. Then,

$V_1=\frac{1}{6}[\vec a\,\,\vec b\,\,\vec c]$

Let $V_2$ be the volume of the parallelopiped with $OG_1, OG_2, OG_3$ as coterminus edges. Then,

$V_2 =[\vec{OG_1}\,\, \vec{OG_2}\,\, \vec{OG_3}]$

$⇒V_2 =\left[\frac{\vec b+\vec c}{3},\frac{\vec c+\vec a}{3},\frac{\vec a+\vec b}{3}\right]$

$⇒V_2 =\frac{1}{27}\begin{bmatrix}\vec b+\vec c&\vec c+\vec a&\vec a+\vec b\end{bmatrix}=\frac{2}{27}[\vec a\,\,\vec b\,\,\vec c]$

$∴\frac{V_1}{V_2}=\frac{\frac{1}{6}[\vec a\,\,\vec b\,\,\vec c]}{\frac{2}{27}[\vec a\,\,\vec b\,\,\vec c]}=\frac{9}{4}$

Hence, $V_1:V_2=9:4$

So, statement-1 is true.

Also, statement-2 is a correct explanation of statement-1.