If the three coterminous edges of a parallelepiped are represented by the vectors $(a - b)$,$(b - c)$,and $(c - a)$,find its volume.

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

If the vectors $\overrightarrow{a}=\hat{i}+a \hat{j}+a^{2} \hat{k}$,$\overrightarrow{b}=\hat{i}+b \hat{j}+b^{2} \hat{k}$ and $\overrightarrow{c}=\hat{i}+c \hat{j}+c^{2} \hat{k}$ are three non-coplanar vectors and $\left|\begin{array}{lll}a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0$,then the value of $abc$ is

If $35 \hat{i}+14 \hat{j}-77 \hat{k}$,$2 \hat{i}+7 \hat{j}+5 \hat{k}$ and $5 \hat{i}+2 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda$ is equal to

If $\bar{a}=2 \hat{i}-\hat{j}+\hat{k}, \bar{b}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\bar{c}=3 \hat{i}+\lambda \hat{j}+5 \hat{k}$ are coplanar,then $\lambda$ is the root of the equation

If the four points with position vectors $-2\hat{i}+\hat{j}+\hat{k}$,$\hat{i}+\hat{j}+\hat{k}$,$\hat{j}-\hat{k}$,and $\lambda\hat{j}+\hat{k}$ are coplanar,then $\lambda$ is equal to