If $a = (1, -1, 1)$ and $c = (-1, -1, 0),$ then the vector $b$ satisfying $a \times b = c$ and $a \cdot b = 1$ is

Let $m$ be a vector of magnitude $\sqrt{3}$ and perpendicular to the vectors $\hat{i}+\hat{j}$ and $\hat{j}-\hat{k}$. Let $n$ be another vector of magnitude $2\sqrt{6}$ and perpendicular to the vectors $2\hat{i}-\hat{j}$ and $\hat{j}+2\hat{k}$. The area (in sq. units) of the triangle formed with $m$ and $n$ as sides is

$\vec{r}$ is a vector perpendicular to the plane determined by the vectors $2 \hat{i}-\hat{j}$ and $\hat{j}+2 \hat{k}$. If the magnitude of the projection of $\vec{r}$ on the vector $2 \hat{i}+\hat{j}+2 \hat{k}$ is $1$, then $|\vec{r}|=$

The area of a parallelogram whose diagonals are the vectors $2 \bar{a}-\bar{b}$ and $4 \bar{a}-5 \bar{b}$,where $\bar{a}$ and $\bar{b}$ are unit vectors forming an angle of $45^{\circ}$ is

If the vertices of a triangle are $A(1, -1, 2)$,$B(2, 0, -1)$,and $C(0, 2, 1)$,then the area of the triangle is:

If $C$ is a given non-zero scalar and $\overline{A}$ and $\overline{B}$ are given non-zero vectors such that $\overline{A}$ is perpendicular to $\overline{B}$. If vector $\overline{X}$ is such that $\overline{A} \cdot \overline{X} = C$ and $\overline{A} \times \overline{X} = \overline{B}$,then $\overline{X}$ is given by: