The area of a parallelogram whose two adjacent sides are represented by the vectors $\vec{a} = 3i - k$ and $\vec{b} = i + 2j$ 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 $|\vec{r}|=$

If $2\vec{a} + 3\vec{b} + \vec{c} = \vec{0}$,then $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$ is equal to

The number of vectors of unit length perpendicular to vectors $a = (1, 1, 0)$ and $b = (0, 1, 1)$ is

For vectors $\vec{a}$ and $\vec{b}$,if $|\vec{a}|=3$,$|\vec{b}|=\frac{\sqrt{2}}{3}$ and $\vec{a} \times \vec{b}$ is a unit vector,then the angle between the two vectors $\vec{a}$ and $\vec{b}$ is . . . . . . .

Let $\bar{a}$,$\bar{b}$,and $\bar{c}$ be unit vectors. Suppose that $\bar{a} \cdot \bar{b} = \bar{a} \cdot \bar{c} = 0$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{6}$. Then $\bar{a}$ is equal to: