Let the volume of tetrahedron $ABCD$ be $81$ cubic units and $G_1, G_2, G_3$ be the centroids of the triangular faces $ABC, ABD,$ and $ACD$ respectively. Then the volume of tetrahedron $AG_1G_2G_3$ is (in cubic units):

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is

The value of $\lambda$ for which the four points $2i + 3j - k$,$i + 2j + 3k$,$3i + 4j - 2k$,and $i - \lambda j + 6k$ are coplanar is:

The value of $\alpha$,so that the volume of the parallelepiped formed by $\hat{i}+\alpha \hat{j}+\hat{k}$,$\hat{j}+\alpha \hat{k}$,and $\alpha \hat{i}+\hat{k}$ becomes minimum,is

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to

Volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $12$ cubic units. The volume of a tetrahedron with coterminous edges $\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{a} + \vec{b} - \vec{c}$ will be ............. cubic units.