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

If the vectors $\vec{b}, \vec{c}, \vec{d}$ are not coplanar, then the vector $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})+(\vec{a} \times \vec{c}) \times(\vec{d} \times \vec{b})+(\vec{a} \times \vec{d}) \times(\vec{b} \times \vec{c})$ is

If $a = i + j - k$,$b = 2i + 3j + k$ and $c = i + \alpha j$ are coplanar vectors,the value of $\alpha$ is

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} - \hat{j}$ and $\vec{c}$ are three vectors such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$, then $\vec{c} \cdot (\vec{a} - 2\vec{b})$ is equal to . . . . . . .

Consider the set of eight vectors $V=\{a \hat{i}+b \hat{j}+c \hat{k}: a, b, c \in\{-1,1\}\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

Let the vectors $\vec{u} = (2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}$,$\vec{v} = (1+b) \hat{i}+2 b \hat{j}-b \hat{k}$,and $\vec{w} = (2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}$ where $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true?