If $A, B, C$ and $D$ are $(3,7,4), (5,-2,-3), (-4,5,6)$ and $(1,2,3)$ respectively,then the volume of the parallelepiped with $AB, AC$ and $AD$ as the co-terminus edges is .... cubic units.

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

If $\vec{r}$ is a vector perpendicular to both the vectors $2 \hat{i}+3 \hat{j}-4 \hat{k}$ and $3 \hat{i}-\hat{j}+\hat{k}$ and satisfies $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+4 \hat{k})=5$,then $|\vec{r}|=$

If the volume of a tetrahedron whose vertices are $A \equiv (1, -6, 10)$,$B \equiv (-1, -3, 7)$,$C \equiv (5, -1, k)$,and $D \equiv (7, -4, 7)$ is $11$ cubic units,then the value of $k$ is:

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelepiped of volume $V$. Then the volume of the parallelepiped,whose coterminous edges are represented by $\vec{a}, \vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to $..........\,V$.

Value of $\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{i} \cdot(\hat{j} \times \hat{j})+\hat{k} \cdot(\hat{j} \times \hat{i})+\hat{i} \cdot(\hat{k} \times \hat{j})$ is . . . . . . .