Let $\overrightarrow{a}=\hat{i}-2 \hat{j}$,$\overrightarrow{b}=2 \hat{j}+3 \hat{k}$,$\overrightarrow{c}=p\hat{i}+q \hat{j}$ and $\overrightarrow{d}=p \hat{j}-q \hat{k}$ be four vectors. If $(\vec{a} \times \vec{b}) \cdot \vec{c}=3=(\vec{a} \times \vec{b}) \cdot \vec{d}$,then $3 p+q=$

The edges of a parallelepiped are of unit length and are parallel to non-coplanar unit vectors $\hat{a}, \hat{b}, \hat{c}$ such that $\hat{a} \cdot \hat{b} = \hat{b} \cdot \hat{c} = \hat{c} \cdot \hat{a} = 1/2$. Then the volume of the parallelepiped is

If the vector $\overline{c}$ lies in the plane of $\overline{a}$ and $\overline{b}$,where $\overline{a}=\hat{i}-\hat{j}+2\hat{k}$,$\overline{b}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{c}=x\hat{i}-(2-x)\hat{j}-\hat{k}$,then the value of $x$ is

Let $OA, OB, OC$ be the co-terminal edges of a rectangular parallelopiped of volume $V$ and let $P$ be the vertex opposite to $O$. Then, $[\overrightarrow{AP} \overrightarrow{BP} \overrightarrow{CP}]$ is equal to

The volume of the tetrahedron whose vertices are $A(-1, 2, 3)$, $B(3, -2, 1)$, $C(p, 1, 3)$, and $D(-1, -2, 4)$ is $\frac{16}{3}$ cubic units. Find the value of $p$.

Let $\overline{a}, \overline{b}, \overline{c}$ be three non-coplanar vectors and $\overline{p}, \overline{q}, \overline{r}$ be defined by the relations $\overline{p}=\frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, \overline{q}=\frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}, \overline{r}=\frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}$. Then the value of the expression $(\overline{a}+\overline{b}) \cdot \overline{p}+(\overline{b}+\overline{c}) \cdot \overline{q}+(\overline{c}+\overline{a}) \cdot \overline{r}$ is equal to: