Let $\vec V = 2\hat i + \hat j - \hat k$,$\vec W = \hat i + 3\hat k$,and $|\vec U| = 2$. If $\vec U$ is a vector in the $x-y$ plane,then the greatest value of $([\vec U \vec V \vec W])^2$ is:

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then the vectors $a + 2b + 3c, \lambda b + 4c$ and $(2\lambda - 1)c$ are non-coplanar for

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:

If the volume of the parallelepiped formed by three non-coplanar vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $4$ cubic units,then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is equal to

If the volume of the tetrahedron whose coterminous edges are given by the vectors $\vec{a} = -2\hat{i} + 3\hat{j} - 3\hat{k}$, $\vec{b} = 4\hat{i} + 5\hat{j} + (\lambda - 10)\hat{k}$, and $\vec{c} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ is $11$ cubic units, then the sum of the possible values of $\lambda$ is