If the points with position vectors $3i - 2j - k$,$2i + 3j - 4k$,$-i + j + 2k$,and $4i + 5j + \lambda k$ are coplanar,then $\lambda = \dots$

The volume of a parallelepiped whose coterminous edges are represented by the vectors $\vec{OA}, \vec{OB},$ and $\vec{OC}$ with vertices $A(4, 3, 1), B(3, 1, 2),$ and $C(5, 2, 1)$ where $O$ is the origin,is ......... cubic units.

The volume of the parallelepiped whose coterminous edges are represented by the vectors $2i - 3j + 4k$,$i + 2j - 2k$,and $3i - j + k$ is ............ $cubic \ units$.

Find the volume of a tetrahedron whose vertices are given by the vectors $-i + j + k$,$i - j + k$,and $i + j - k$,with the fourth vertex being the origin.

The volume of a parallelepiped whose coterminous edges are $2 \overrightarrow{a}, 2 \overrightarrow{b}, 2 \overrightarrow{c}$ is:

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?