If the vectors $2i - 3j + 4k$, $i + 2j - k$ and $xi - j + 2k$ are coplanar, then $x = $

The volume of the parallelopiped whose coterminous edges are $\hat{j}+\hat{k}$, $\hat{i}+\hat{k}$, and $\hat{i}+\hat{j}$ is

If the vectors $\overrightarrow{a}+\lambda \overrightarrow{b}+3 \overrightarrow{c}$,$-2 \overrightarrow{a}+3 \overrightarrow{b}-4 \overrightarrow{c}$ and $\overrightarrow{a}-3 \overrightarrow{b}+5 \overrightarrow{c}$ are coplanar,then the value of $\lambda$ is

The value of $a$,so that the volume of the parallelepiped formed by $\hat{i} + a \hat{j} + \hat{k}$,$\hat{j} + a \hat{k}$,and $a \hat{i} + \hat{k}$ becomes minimum is

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to

If $a, b, c$ are non-coplanar vectors and $d = \lambda a + \mu b + \nu c$,then $\lambda$ is equal to