If the vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$,$\vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}$,and $\vec{c}=3 \hat{i}+p \hat{j}+5 \hat{k}$ are coplanar,then $p=$

If $\overline{a}, \overline{b}$ and $\overline{c}$ are three non-coplanar vectors,then $(\overline{a}+\overline{b}+\overline{c}) \cdot[(\overline{a}+\overline{b}) \times(\overline{a}+\overline{c})]$ equals

For any non-zero vectors $\bar{a}$ and $\bar{b}$,$\left[\begin{array}{lll}\bar{b} & \bar{a} \times \bar{b} & \bar{a}\end{array}\right]=$

Let the volume of the tetrahedron with vertices $\hat{i}-\hat{j}-2\hat{k}$,$-2\hat{i}+\hat{j}-2\hat{k}$,$-\hat{i}-2\hat{j}+\hat{k}$,and $2\hat{i}+2\hat{j}+a\hat{k}$ be $\frac{20}{3}$. Then the integral value of $a$ is

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 $\vec{p} = 2\hat{i} + 3\hat{j} + a\hat{k}$,$\vec{q} = b\hat{i} + 5\hat{j} - \hat{k}$,and $\vec{r} = \hat{i} + \hat{j} + 3\hat{k}$. If $\vec{p}, \vec{q}, \vec{r}$ are coplanar and $\vec{p} \cdot \vec{q} = 20$,then the ordered pair $(a, b)$ is: