If $[(\overline{a}+2 \overline{b}+3 \overline{c}) \times(\overline{b}+2 \overline{c}+3 \overline{a})] \cdot(\overline{c}+2 \overline{a}+3 \overline{b})=54$,then the value of $[\overline{a} \ \overline{b} \ \overline{c}]$ is

Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{a} \times \vec{b} = \vec{c}$,$\vec{b} \times \vec{c} = \vec{a}$ and $|\vec{a}| = 2$. Then which one of the following is not true?

The volume of the parallelepiped whose conterminous edges are $\vec{a} = \hat{i} - \hat{j} + \hat{k}$,$\vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k}$,and $\vec{c} = 3\hat{i} - 5\hat{j} + 2\hat{k}$ is:

$\bar{a} = \hat{i} + \hat{j} + \hat{k}$,$\bar{b} = \hat{i} - \hat{j} + 2\hat{k}$ and $\bar{c} = x\hat{i} + (x-1)\hat{j} - \hat{k}$. If the vector $\bar{c}$ lies in the plane of $\bar{a}$ and $\bar{b}$,then $x=$

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and $\bar{p}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{q}=\frac{\bar{c} \times \bar{a}}{[\bar{a} \bar{b} \bar{c}]}, \bar{r}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}$,then $\bar{a} \cdot \bar{p}+\bar{b} \cdot \bar{q}+\bar{c} \cdot \bar{r}=$

For any non-zero vectors $\bar{a}$ and $\bar{b}$,$\left[\begin{array}{lll}\bar{b} & \bar{a} \times \bar{b} & \bar{a}\end{array}\right]=$