If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

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 the volume of the parallelepiped with $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ as coterminous edges is $40 \text{ cubic units}$,then the volume of the parallelepiped having $\overrightarrow{b}+\overrightarrow{c}, \overrightarrow{c}+\overrightarrow{a}$ and $\overrightarrow{a}+\overrightarrow{b}$ as coterminous edges in cubic units is

If $\bar{a}$ is perpendicular to $\bar{b}$ and $\bar{c}$,$|\vec{a}|=2$,$|\bar{b}|=3$,$|\bar{c}|=4$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{3}$,then $\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]=$ (in $\sqrt{3}$)

If $a \cdot b = b \cdot c = c \cdot a = 0$,then what is the value of the scalar triple product $[a b c]$?

The maximum value and minimum value of the volume of the parallelepiped having coterminous edges $\hat{i}+x \hat{j}+\hat{k}$,$\hat{j}+x \hat{k}$,and $x \hat{i}+\hat{k}$ are respectively: