The volume of a parallelepiped whose coterminous edges are represented by the vectors $\overrightarrow{OA} = (2, 1, 1)$,$\overrightarrow{OB} = (3, -1, 1)$,and $\overrightarrow{OC} = (-1, 1, -1)$ is . . . . . . cubic units.

If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + aj + 5k$ are coplanar,find the value of $a$.

Unit vectors $a, b$ and $c$ are coplanar. $A$ unit vector $d$ is perpendicular to them. 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 $c$ is:

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelepiped of volume $V$. Then the volume of the parallelepiped,whose coterminous edges are represented by $\vec{a}, \vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to $..........\,V$.

If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$,and $\hat{i}+\hat{j}+c \hat{k}$ are coplanar,where $(a, b, c \neq 1)$,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{r}$ is any vector,then $[\vec{b} \, \vec{c} \, \vec{r}] \vec{a} + [\vec{c} \, \vec{a} \, \vec{r}] \vec{b} + [\vec{a} \, \vec{b} \, \vec{r}] \vec{c} = \dots$