If $a, b, c$ are any three non-coplanar vectors,then $[a + b, b + c, c + a] = $

If $\vec{a}$ and $\vec{b}$ are mutually perpendicular unit vectors and $\vec{r}$ is a vector such that $\vec{r} \cdot \vec{a} = 0$,$\vec{r} \cdot \vec{b} = 1$,and $[\vec{r} \, \vec{a} \, \vec{b}] = 1$,then $\vec{r} = \dots$

If the volume of a parallelepiped whose coterminous edges are represented by the vectors $-12i + \alpha k$,$3j - k$,and $2i + j - 15k$ is $546$,find the value of $\alpha$.

Consider the set of eight vectors $V=\{a \hat{i}+b \hat{j}+c \hat{k}: a, b, c \in\{-1,1\}\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

The vectors $i + 2j + 3k$,$\lambda i + 4j + 7k$,and $-3i - 2j - 5k$ are collinear if $\lambda$ equals:

Let $a=\hat{i}+2 \hat{j}-\hat{k}$ and $b=\hat{i}+\hat{j}+\hat{k}$. If $p$ is a unit vector such that $[a b p]$ is maximum, then $p=$