For any non-zero vectors $\bar{a}, \bar{b}, \bar{c}$,the value of $\bar{a} \cdot [(\bar{b} \times \bar{c}) \times (\bar{a} + \bar{b} + \bar{c})]$ is

If $u, v$ and $w$ are three non-coplanar vectors,then $(u + v - w) \cdot [(u - v) \times (v - w)]$ equals

If $a, b, c$ are three non-coplanar vectors and $p, q, r$ are defined by the relations $p = \frac{b \times c}{[a, b, c]}, q = \frac{c \times a}{[a, b, c]}, r = \frac{a \times b}{[a, b, c]}$,then $(a+b) \cdot p + (b+c) \cdot q + (c+a) \cdot r =$

If $a, b$ and $c$ are non-coplanar,then the value of $a \cdot \left\{ \frac{b \times c}{3 b \cdot (c \times a)} \right\} - b \cdot \left\{ \frac{c \times a}{2 c \cdot (a \times b)} \right\}$ 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 the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$, $2 \bar{i}+\bar{j}-3 \bar{k}$, and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$, then the values of $p$ are the roots of the equation