If the points $A(1,1,2), B(2,1, p), C(1,0,3)$ and $D(2,2,0)$ are coplanar,then the value of $p$ is

Let $a=p(\hat{i}+\hat{j}+\hat{k})$, $b=\hat{i}+\hat{j}-2\hat{k}$, and $c=2\hat{i}-\hat{j}+2\hat{k}$ be three vectors. If the value of $[abc]$ is not more than $15$ and not less than $-5$, then $p$ lies in the interval:

If $\overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} + 3 \hat{k}$,$\overrightarrow{b} = -\beta \hat{i} - \alpha \hat{j} - \hat{k}$,and $\overrightarrow{c} = \hat{i} - 2 \hat{j} - \hat{k}$ such that $\overrightarrow{a} \cdot \overrightarrow{b} = 1$ and $\overrightarrow{b} \cdot \overrightarrow{c} = -3$,then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ 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 $..........\,V$.

If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

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 . . . . . . .