If the volume of a tetrahedron whose vertices are $A \equiv (1, -6, 10)$,$B \equiv (-1, -3, 7)$,$C \equiv (5, -1, k)$,and $D \equiv (7, -4, 7)$ is $11$ cubic units,then the value of $k$ is:

If $a, b$ and $c$ are three non-coplanar vectors,then $(a + b + c) \cdot [(a + b) \times (a + c)]$ is equal to

If $a = i + j - k$,$b = 2i + 3j + k$ and $c = i + \alpha j$ are coplanar vectors,the value of $\alpha$ is

If the three coterminous edges of a parallelepiped are represented by the vectors $(a - b)$,$(b - c)$,and $(c - a)$,find its volume.

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

The sum of all values of $\alpha$,for which the points whose position vectors $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}-3 \hat{j}+4 \hat{k}$,$(\alpha+1) \hat{i}+2 \hat{k}$ and $9 \hat{i}+(\alpha-8) \hat{j}+6 \hat{k}$ are coplanar,is equal to