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 $\bar{a}$ is perpendicular to $\bar{b}$ and $\bar{c}$,$|\vec{a}|=2$,$|\bar{b}|=3$,$|\bar{c}|=4$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{3}$,then $\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]=$ (in $\sqrt{3}$)

$[(a \times b) \times (b \times c), (b \times c) \times (c \times a), (c \times a) \times (a \times b)] = \,$

If the volume of a tetrahedron,whose vertices are $A(1, 2, 3)$,$B(-3, -1, 1)$,$C(2, 1, 3)$,and $D(-1, 2, x)$ is $\frac{11}{6}$ cubic units,then the value of $x$ is:

If the volume of the parallelepiped formed by the vectors $\hat{i} + \lambda \hat{j} + \hat{k}$,$\hat{j} + \lambda \hat{k}$ and $\lambda \hat{i} + \hat{k}$ is minimum,then $\lambda$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $(\vec{a} - \lambda \vec{b}) \cdot (\vec{b} - 2\vec{c}) \times (\vec{c} + 2\vec{a}) = 0$,then $\lambda$ is equal to