If $a, b$ and $c$ are three non-coplanar vectors,then $(a + b + c) \cdot [(a + b) \times (a + c)]$ is equal to

$(\hat{i} \times \hat{j}) \cdot [(\hat{j} \times \hat{k}) \times (\hat{k} \times \hat{i})]$

If the vectors $\vec{a} = \hat{i} + a\hat{j} + \hat{k}$,$\vec{b} = \hat{j} + a\hat{k}$,and $\vec{c} = a\hat{i} + \hat{k}$ are given,find the value of $a$ for which the volume of the parallelepiped formed by these three vectors as coterminous edges is minimum.

Let $\vec{u} = a\hat{i} + b\hat{j} + c\hat{k}$,$\vec{v} = b\hat{i} + c\hat{j} + a\hat{k}$,and $\vec{w} = c\hat{i} + a\hat{j} + b\hat{k}$. If $[\vec{u} \, \vec{v} \, \vec{w}] = 0$ and $\vec{w} = \lambda \vec{x} + \mu \vec{y}$ where $(a + b + c) \neq 0$ and $\lambda, \mu \neq 0$,then the vectors $\vec{x}, \vec{y}, \vec{u}, \vec{v}, \vec{w}$ are:

If the vectors $\hat{\imath}+\hat{\jmath}+\hat{k}$,$\hat{\imath}-\hat{\jmath}+\hat{k}$ and $2\hat{\imath}+3\hat{\jmath}+m\hat{k}$ are coplanar,then $m=$

If the vectors $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar,then $\frac{[\bar{a}+2\bar{b} \quad \bar{b}+2\bar{c} \quad \bar{c}+2\bar{a}]}{[\bar{a} \quad \bar{b} \quad \bar{c}]}=$