Let $a, b$ and $c$ be three non-coplanar vectors and let $p, q$ and $r$ be the vectors defined by $p=\frac{b \times c}{[a b c]}, q=\frac{c \times a}{[a b c]}, r=\frac{a \times b}{[a b c]}$. Then, $(a+b) \cdot p+(b+c) \cdot q+(c+a) \cdot r$ is equal to

The volume of the parallelepiped determined by the vectors $\vec{a} + \vec{b}, \vec{b} + \vec{c}$ and $\vec{c} + \vec{a}$ is $4$. Then the volume of the parallelepiped determined by the vectors $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ is:

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 $p=$

If $\alpha = 2i + 3j - k$,$\beta = -i + 2j - 4k$,and $\gamma = i + j + k$,then $(\alpha \times \beta) \cdot (\alpha \times \gamma)$ is equal to

If $a$ and $b$ are unit vectors perpendicular to each other, then $[a + (a \times b) \quad b + (a \times b) \quad (a \times b)]$ =

If $[\bar{a} \quad \bar{b} \quad \bar{c}]=4$,then the volume of the parallelepiped with coterminous edges $\bar{a}+2 \bar{b}$,$\bar{b}+2 \bar{c}$,and $\bar{c}+2 \bar{a}$ is: