If $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + 3\hat{j} - \hat{k}$,and $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ are coplanar vectors,then $\lambda = . . . .$

Let $\overline{a} = \hat{i} + \hat{j} + \hat{k}$,$\overline{b} = \hat{i} - \hat{j} + 2\hat{k}$,and $\overline{c} = x\hat{i} + (x - 2)\hat{j} - \hat{k}$. If the vector $\overline{c}$ lies in the plane of $\overline{a}$ and $\overline{b}$,then $x = \dots$

If $3 \hat{i}+3 \hat{j}+\sqrt{3} \hat{k}$,$\hat{i}+\hat{k}$,and $\sqrt{3} \hat{i}+\sqrt{3} \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda$ is equal to

For three vectors $u, v, w$,which of the following expressions is not equal to any of the remaining three?

Let $\overrightarrow{PR}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $\overrightarrow{SQ}=\hat{i}-3 \hat{j}-4 \hat{k}$ be the diagonals of a parallelogram $PQRS$,and let $\overrightarrow{PT}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{PT}, \overrightarrow{PQ}$ and $\overrightarrow{PS}$ is:

If the points $A(3,2,1)$,$B(4, x, 5)$,$C(4,2,-2)$,and $D(6,5,-1)$ are coplanar,then $x$ has the value: