If the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$,$2 \bar{i}+\bar{j}-3 \bar{k}$,and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$,then the values of $p$ are the roots of the equation

If $[\bar{a} \quad \bar{b} \quad \bar{c}]=4$,then the volume of the parallelepiped with coterminous edges $\bar{a}+2 \bar{b}$,$\bar{b}+2 \bar{c}$,and $\bar{c}+2 \bar{a}$ is:

Let $\overrightarrow{OP} = \frac{\alpha-1}{\alpha} \hat{i} + \hat{j} + \hat{k}$,$\overrightarrow{OQ} = \hat{i} + \frac{\beta-1}{\beta} \hat{j} + \hat{k}$ and $\overrightarrow{OR} = \hat{i} + \hat{j} + \frac{1}{2} \hat{k}$ be three vectors,where $\alpha, \beta \in \mathbb{R} - \{0\}$ and $O$ denotes the origin. If $(\overrightarrow{OP} \times \overrightarrow{OQ}) \cdot \overrightarrow{OR} = 0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3x + 3y - z + l = 0$,then the value of $l$ is:

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then the vectors $a + 2b + 3c, \lambda b + 4c$ and $(2\lambda - 1)c$ are non-coplanar for

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

If $a, b, c$ are three non-coplanar vectors,then $\frac{a \cdot (b \times c)}{c \times a \cdot b} + \frac{b \cdot (a \times c)}{c \cdot (a \times b)} = $