The value of $x$ for which the volume of the parallelepiped formed by the vectors $\vec{a} = \hat{i} + x\hat{j} + \hat{k}$, $\vec{b} = \hat{j} + x\hat{k}$, and $\vec{c} = x\hat{i} + \hat{k}$ is minimum, is

If $\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$,and $\bar{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$,and $[3 \bar{a}+\bar{b} \quad 3 \bar{b}+\bar{c} \quad 3 \bar{c}+\bar{a}] = \lambda \begin{vmatrix} \bar{a} \cdot \hat{i} & \bar{a} \cdot \hat{j} & \bar{a} \cdot \hat{k} \\ \bar{b} \cdot \hat{i} & \bar{b} \cdot \hat{j} & \bar{b} \cdot \hat{k} \\ \bar{c} \cdot \hat{i} & \bar{c} \cdot \hat{j} & \bar{c} \cdot \hat{k} \end{vmatrix}$,then the value of $\lambda$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors,then $\left| {\begin{array}{*{20}{c}} {\vec{a} \cdot \vec{a}} & {\vec{a} \cdot \vec{b}} & {\vec{a} \cdot \vec{c}} \\ {\vec{b} \cdot \vec{a}} & {\vec{b} \cdot \vec{b}} & {\vec{b} \cdot \vec{c}} \\ {\vec{c} \cdot \vec{a}} & {\vec{c} \cdot \vec{b}} & {\vec{c} \cdot \vec{c}} \end{array}} \right| = \dots$

The volume of the tetrahedron having the edges $\hat{i}+2\hat{j}-\hat{k}$, $\hat{i}+\hat{j}+\hat{k}$, and $\hat{i}-\hat{j}+\lambda\hat{k}$ as coterminous edges is $\frac{2}{3}$ cubic units. Then $\lambda$ equals:

Statement-$1$: If the points $(1, 2, 2), (2, 1, 2), (2, 2, z)$ and $(1, 1, 1)$ are coplanar,then $z = 2$.
Statement-$2$: If $4$ points $P, Q, R$ and $S$ are coplanar,then the volume of the tetrahedron $PQRS$ is $0$.

Consider the vectors $\vec{x}=\hat{i}+2\hat{j}+3\hat{k}$,$\vec{y}=2\hat{i}+3\hat{j}+\hat{k}$,and $\vec{z}=3\hat{i}+\hat{j}+2\hat{k}$. For two distinct positive real numbers $\alpha$ and $\beta$,define $\vec{X}=\alpha\vec{x}+\beta\vec{y}-\vec{z}$,$\vec{Y}=\alpha\vec{y}+\beta\vec{z}-\vec{x}$,and $\vec{Z}=\alpha\vec{z}+\beta\vec{x}-\vec{y}$. If the vectors $\vec{X}, \vec{Y}$,and $\vec{Z}$ lie in a plane,the value of $\alpha+\beta-3$ is $....$.