If $a = 2i + 3j - 5k$,$b = mi + nj + 12k$ and $a \times b = 0$,then $(m, n) = $

$A$ unit vector perpendicular to the plane determined by the points $P(1, -1, 2)$,$Q(2, 0, -1)$,and $R(0, 2, 1)$ is:

If $A(3, 1, -1)$,$B\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right)$,$C(2, 2, 1)$ and $D\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral $ABCD$,then its area is

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have the same magnitude. If a vector $\vec{r}$ satisfies $\vec{a} \times \{(\vec{r}-\vec{b}) \times \vec{a}\} + \vec{b} \times \{(\vec{r}-\vec{c}) \times \vec{b}\} + \vec{c} \times \{(\vec{r}-\vec{a}) \times \vec{c}\} = \vec{0}$,then $\vec{r}$ is equal to:

Find a unit vector coplanar with $i + 2j + k$ and $i + j + 2k$ and perpendicular to $i + j + k$.

Let $\overrightarrow{c}$ be a vector perpendicular to the vectors $\overrightarrow{a}=\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}.$ If $\overrightarrow{c}\cdot(\hat{i}+\hat{j}+3\hat{k})=8,$ then the value of $\overrightarrow{c}\cdot(\overrightarrow{a}\times\overrightarrow{b})$ is equal to ...... .