If the vectors $2\hat{i}-\hat{j}+3\hat{k}$,$\hat{i}+4\hat{j}+\hat{k}$,and $4\hat{i}+p\hat{j}+\hat{k}$ are coplanar,then $p=$

Statement $1$: The vectors $\vec{a}, \vec{b}$ and $\vec{c}$ lie in the same plane if and only if $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$.
Statement $2$: The vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$,where $\vec{u} \times \vec{v}$ is a vector perpendicular to the plane of $\vec{u}$ and $\vec{v}$.

If $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular vectors such that $|\bar{a}| = a, |\bar{b}| = b, |\bar{c}| = c$,then $[\bar{a} \bar{b} \bar{c}] = ......$

Let the vectors $\vec{u} = (2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k}$,$\vec{v} = (1+b) \hat{i}+2 b \hat{j}-b \hat{k}$,and $\vec{w} = (2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}$ where $a, b, c \in \mathbb{R}$ be co-planar. Then which of the following is true?

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.

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are non-zero coplanar vectors,then $[2 \overrightarrow{a}-\overrightarrow{b} \quad 3 \overrightarrow{b}-\overrightarrow{c} \quad 4 \overrightarrow{c}-\overrightarrow{a}]$ is