If the volume of a parallelepiped whose coterminous edges are represented by the vectors $-12i + \alpha k$,$3j - k$,and $2i + j - 15k$ is $546$,find the value of $\alpha$.

If the points having the position vectors $-\hat{i}+4 \hat{j}-4 \hat{k}$,$3 \hat{i}+2 \hat{j}-5 \hat{k}$,$-3 \hat{i}+8 \hat{j}-5 \hat{k}$ and $-3 \hat{i}+2 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda=$

If $\vec{OA}=6 \hat{i}+3 \hat{j}-4 \hat{k}$, $\vec{OB}=2 \hat{j}+\hat{k}$, and $\vec{OC}=5 \hat{i}-\hat{j}+2 \hat{k}$ are the coterminous edges of a parallelepiped, then the height of the parallelepiped drawn from the vertex $A$ is

If $A, B, C$ and $D$ are $(3,7,4), (5,-2,-3), (-4,5,6)$ and $(1,2,3)$ respectively,then the volume of the parallelepiped with $AB, AC$ and $AD$ as the co-terminus edges is .... cubic units.

If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$,and $\hat{i}+\hat{j}+c \hat{k}$ are coplanar,where $(a, b, c \neq 1)$,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number,then for what value of $\lambda$ are the vectors $\vec{a} + 2\vec{b} + 3\vec{c}$,$\lambda\vec{b} + 4\vec{c}$,and $(2\lambda - 1)\vec{c}$ non-coplanar?