If the vectors $a$,$b$,and $c$ are coplanar,then $\left|\begin{array}{ccc}a & b & c \\ a \cdot a & a \cdot b & a \cdot c \\ b \cdot a & b \cdot b & b \cdot c\end{array}\right|$ is equal to

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are non-zero coplanar vectors,then $[2 \overrightarrow{a}-\overrightarrow{b} \quad 3 \overrightarrow{b}-\overrightarrow{c} \quad 4 \overrightarrow{c}-\overrightarrow{a}]$ is

If $\bar{a}, \bar{b},$ and $\bar{c}$ are mutually perpendicular vectors such that $|\bar{a}| = 1, |\bar{b}| = 3,$ and $|\bar{c}| = 5,$ then find the value of $[\bar{a} - 2\bar{b}, \bar{b} - 3\bar{c}, \bar{c} - 4\bar{a}]$.

The volume of a parallelepiped whose coterminous edges are represented by the vectors $\vec{OA}, \vec{OB},$ and $\vec{OC}$ with vertices $A(4, 3, 1), B(3, 1, 2),$ and $C(5, 2, 1)$ where $O$ is the origin,is ......... cubic units.

If $[(\overline{a}+2 \overline{b}+3 \overline{c}) \times(\overline{b}+2 \overline{c}+3 \overline{a})] \cdot(\overline{c}+2 \overline{a}+3 \overline{b})=54$,then the value of $[\overline{a} \ \overline{b} \ \overline{c}]$ is

If $a = i + j + k$,$b = 4i + 3j + 4k$ and $c = i + \alpha j + \beta k$ are linearly dependent vectors and $|c| = \sqrt{3}$,then