The volume of the parallelepiped formed by the vectors $\hat{i} + m \hat{j} + \hat{k}$,$\hat{j} + m \hat{k}$,and $m \hat{i} + \hat{k}$ becomes minimum when $m$ is

$a \cdot [(b + c) \times (a + b + c)]$ is equal to

If $a$ and $b$ are parallel vectors,then $[a \ c \ b] = $

If the origin and the points $(1, 2, 3)$,$(2, 3, 4)$,and $(x, y, z)$ are coplanar,then

If $a, b$ and $c$ are three non-coplanar vectors and $p, q$ and $r$ are 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

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