Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{j}-\hat{k}.$ If $\overrightarrow{c}$ is a vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$,then $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal to :

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 $..........\,V$.

$i \cdot(j \times k)+j \cdot(k \times i)+k \cdot(j \times i)$ is equal to

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

Let $\overrightarrow{PR}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $\overrightarrow{SQ}=\hat{i}-3 \hat{j}-4 \hat{k}$ be the diagonals of a parallelogram $PQRS$,and let $\overrightarrow{PT}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{PT}, \overrightarrow{PQ}$ and $\overrightarrow{PS}$ is:

If the vertices of a tetrahedron are $\vec{a} = \vec{j} + 2\vec{k}$,$\vec{b} = 3\vec{i} + \vec{k}$,$\vec{c} = 4\vec{i} + 3\vec{j} + 6\vec{k}$,and $\vec{d} = 2\vec{i} + 3\vec{j} + 2\vec{k}$,find its volume.