If $a = 2i + j - k$,$b = i + 2j + k$,and $c = i - j + 2k$,then $a \cdot (b \times c) = \dots$

If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$,and $\hat{i}+\hat{j}+c \hat{k}$ are coplanar,where $(a, b, c \neq 1)$,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

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

$(\vec{a}+2 \vec{b}-\vec{c}) \cdot \{(\vec{a}-\vec{b}) \times (\vec{a}-\vec{b}-\vec{c})\} =$

If $a = i + j + k$,$b = 4i + 3j + 4k$,and $c = i + \alpha j + \beta k$ are coplanar vectors and $|c| = \sqrt{3}$,then:

If $\vec{p}$ and $\vec{q}$ are unequal unit vectors such that $(\vec{p} - \vec{q}) \cdot ((2\vec{q} + \vec{p}) \times (3\vec{p} - \vec{q})) = |\vec{p} + \vec{q}|$,then the angle between $\vec{p}$ and $\vec{q}$ is: