For any non-zero vectors $\bar{a}, \bar{b}, \bar{c}$,the value of $\bar{a} \cdot [(\bar{b} \times \bar{c}) \times (\bar{a} + \bar{b} + \bar{c})]$ is

$\vec{a}=2 \hat{i}-\hat{j}$, $\vec{b}=2 \hat{j}-\hat{k}$, $\vec{c}=2 \hat{k}-\hat{i}$ are three vectors and $\vec{d}$ is a unit vector perpendicular to $\vec{c}$. If $\vec{a}, \vec{b}, \vec{d}$ are coplanar vectors, then $|\vec{d} \cdot \vec{b}|=$

If $\vec a = 3\vec j + 4\vec k$,$\vec b = 2\vec i + \vec k$ and $\vec c$,$\vec d$ are respectively the components of $\vec a$ parallel and perpendicular to $\vec b$,then the value of the scalar triple product $\left[ {(\vec a \times \vec c) \times (\vec c \times \vec d), (\vec c \times \vec d) \times (\vec d \times \vec a), (\vec d \times \vec a) \times (\vec a \times \vec c)} \right]$ is equal to:

The volume of the parallelepiped whose coterminous edges are represented by the vectors $2i - 3j + 4k$,$i + 2j - 2k$,and $3i - j + k$ is ............ $cubic \ units$.

The volume of the tetrahedron whose co-terminus edges are $\bar{a}, \bar{b}, \bar{c}$ is $12$ cubic units. If the scalar projection of $\bar{a}$ on $\bar{b} \times \bar{c}$ is $4$,then $|\bar{b} \times \bar{c}|=$

If $a, b,$ and $c$ are unit coplanar vectors,then the scalar triple product $[a, b, c]$ is equal to: