If $a, b, c$ are any three coplanar unit vectors,then

If $a=\hat{i}-2 \hat{j}-3 \hat{k}, b=2 \hat{i}+\hat{j}-\hat{k}, c=\hat{i}+3 \hat{j}-2 \hat{k}$,then $[(a \times b) \times(b \times c), (b \times c) \times(c \times a), (c \times a) \times(a \times b)] = $

If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and $\bar{c}=7 \hat{\imath}-\hat{\jmath}+23 \hat{k}$ are three vectors,then which of the following statements is true?

If $\vec{a} = 4\hat{i} - 2\hat{j} + \hat{k}$,$\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$,and $\vec{c} = 2\hat{i} - \hat{j} + 2\hat{k}$ represent the three coterminous edges of a parallelepiped,find its volume.

If $x$ is parallel to $y$ and $z$ where $x = 2i + j + \alpha k$,$y = \alpha i + k$ and $z = 5i - j$,then $\alpha$ is equal to

If $[\bar{a} \bar{b} \bar{c}] \neq 0$,then $\frac{[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}]}{[\bar{b} \bar{c} \bar{a}]}=$