The volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $3 \text{ cubic units}$. The volume (in cubic units) of a tetrahedron with coterminous edges $(\vec{a} \times \vec{b}), (\vec{a} \times 2\vec{c}), (\vec{b} \times 2\vec{c})$ is:

For three vectors $u, v, w$,which of the following expressions is not equal to any of the remaining three?

If the volume of the parallelepiped with $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ as coterminous edges is $40 \text{ cubic units}$,then the volume of the parallelepiped having $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$ as coterminous edges in cubic units is

The volume of the parallelepiped whose conterminous edges are $\vec{a} = \hat{i} - \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$,and $\vec{c} = 3\hat{i} - 5\hat{j} + 2\hat{k}$ is:

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):

If $A$,$B$,and $C$ are three non-coplanar vectors,then $(A + B + C) \cdot ((A + B) \times (A + C)) = \dots$