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)] = $

Let $a=\hat{i}+\hat{j}$, $b=\hat{j}+\hat{k}$ and $c=\hat{i}+\hat{k}$. If $d$ is a unit vector such that $a \cdot d=0$ and $b \cdot(c \times d)=0$, then $d=$

If the vectors $i+3j-2k$,$2i-j+4k$ and $3i+2j+xk$ are coplanar,then the value of $x$ is:

Let $\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}$,$\vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$,and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha > 0$. If the minimum value of the scalar triple product $[\vec{u} \vec{v} \vec{w}]$ is $-\alpha \sqrt{3401}$,and $|\vec{u} \cdot \hat{i}|^2=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers,then $m + n$ is equal to $.........$.

$[b \times c, c \times a, a \times b]$ is equal to

If $A \equiv (1, -1, 0)$,$B \equiv (0, 1, -1)$,and $C \equiv (-1, 0, 1)$,then the unit vector $\overline{d}$ such that $\overline{a}$ and $\overline{d}$ are perpendicular and $\overline{b}, \overline{c}, \overline{d}$ are coplanar is