$[(a \times b) \times (b \times c), (b \times c) \times (c \times a), (c \times a) \times (a \times b)] = \,$

If $\overrightarrow{a} = 2\hat{i} + \hat{j} + 3\hat{k}$,$\overrightarrow{b} = 3\hat{i} + 3\hat{j} + \hat{k}$ and $\overrightarrow{c} = c_{1}\hat{i} + c_{2}\hat{j} + c_{3}\hat{k}$ are coplanar vectors and $\overrightarrow{a} \cdot \overrightarrow{c} = 5$,$\overrightarrow{b} \perp \overrightarrow{c}$,then $122(c_{1} + c_{2} + c_{3})$ is equal to.......

$[i, k, j] + [k, j, i] + [j, k, i]$

The vectors $\overline{p}=\hat{i}+a \hat{j}+a^2 \hat{k}$,$\overline{q}=\hat{i}+b \hat{j}+b^2 \hat{k}$ and $\overline{r}=\hat{i}+c \hat{j}+c^2 \hat{k}$ are non-coplanar 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 $\overline{a}, \overline{b}, \overline{c}$ are mutually perpendicular vectors having magnitudes $1, 2, 3$ respectively,then the value of $[\overline{a}+\overline{b}+\overline{c} \quad \overline{b}-\overline{a} \quad \overline{c}]$ is

If $[a, b, c] = 3$, then the volume (in cubic units) of the parallelepiped with $2a+b$, $2b+c$, and $2c+a$ as edges is: