The vectors $2 \hat{i}-3 \hat{j}+\hat{k}, \hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+\hat{j}-2 \hat{k}$

If the vectors $p \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+q \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r \hat{k}$ $(p \neq q \neq r \neq 1)$ are coplanar,then the value of $pqr-(p+q+r)$ is

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 $0$.

Unit vectors $a, b, c$ are coplanar. $A$ unit vector $d$ is perpendicular to the given vectors. If $(a \times b) \times (c \times d) = \frac{1}{6}i - \frac{1}{3}j + \frac{1}{3}k$ and the angle between $a$ and $b$ is $30^{\circ}$,then $c = ....$

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

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is