The value of $\lambda$ for which points $A(2, 2, 1)$,$B(1, 1, 1)$,$C(-\lambda, 2, 1)$,and $D(3, 0, -1)$ are coplanar is $\lambda = $ ............

The volume of a parallelepiped whose coterminous edges are represented by the vectors $\overrightarrow{OA} = (2, 1, 1)$,$\overrightarrow{OB} = (3, -1, 1)$,and $\overrightarrow{OC} = (-1, 1, -1)$ is . . . . . . cubic units.

If $\vec{a}=\alpha \hat{i}+\beta \hat{j}+3 \hat{k}$,$\vec{b}=\hat{j}+2 \hat{k}$,and $\vec{c}=3 \hat{i}+2 \hat{j}+\hat{k}$ are linearly dependent vectors and the magnitude of $\vec{a}$ is $\sqrt{14}$. If $\alpha$ and $\beta$ are integers,then $\alpha+\beta=$

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors,then $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})} = \dots$

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 the vectors $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar,then $\frac{[\bar{a}+2\bar{b} \quad \bar{b}+2\bar{c} \quad \bar{c}+2\bar{a}]}{[\bar{a} \quad \bar{b} \quad \bar{c}]}=$