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 $\vec{a}=2 \hat{i}-\hat{j}+3 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}+\hat{k}$,and $\vec{c}=3 \hat{i}-\hat{j}+2 \hat{k}$,then $\vec{a} \cdot(\vec{b} \times \vec{c})=$ . . . . . . .

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

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and the points $P_1 = \lambda \vec{a}+3 \vec{b}-\vec{c}$, $P_2 = \vec{a}-\lambda \vec{b}+3 \vec{c}$, $P_3 = 3 \vec{a}+4 \vec{b}-\lambda \vec{c}$, and $P_4 = \vec{a}-6 \vec{b}+6 \vec{c}$ are coplanar, then one of the values of $\lambda$ is

If the vectors $2 \bar{i} + 4 \bar{j} - 3 \bar{k}$,$-\bar{i} + 2 \bar{j} + 3 \bar{k}$,and $p \bar{i} - 2 \bar{j} + \bar{k}$ are coplanar,then the unit vector in the direction of the vector $9p \bar{i} - 4 \bar{j} + 4 \bar{k}$ is

If $A(-1, 2, 3)$,$B(3, -2, 1)$,$C(2, 1, 3)$ and $D(-1, -2, 4)$ are the vertices of a tetrahedron,then its volume is