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

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

If the four points $2\vec{a} + 3\vec{b} - \vec{c}$,$\vec{a} - 2\vec{b} + 3\vec{c}$,$3\vec{a} + 4\vec{b} - 2\vec{c}$,and $\vec{a} - \lambda\vec{b} - 6\vec{c}$ are coplanar,find the value of $\lambda$.

For three vectors $a, b, c$,the value of $[a \times b, b \times c, c \times a]$ is equal to:

If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(2 \bar{a}-\bar{b}) \cdot [(\bar{a} \times \bar{b}) \times (\bar{a}+2 \bar{b})] = $

Let the vectors $\overrightarrow{u}_1 = \hat{i} + \hat{j} + a\hat{k}$,$\overrightarrow{u}_2 = \hat{i} + b\hat{j} + \hat{k}$ and $\overrightarrow{u}_3 = c\hat{i} + \hat{j} + \hat{k}$ be coplanar. If the vectors $\overrightarrow{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$,$\overrightarrow{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$ and $\overrightarrow{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$ are also coplanar,then $6(a+b+c)$ is equal to $..............$.