Find a vector of magnitude $6,$ which is perpendicular to both the vectors $\vec{a} = 2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b} = 4 \hat{i}-\hat{j}+3 \hat{k}$.

If the area of the parallelogram with $\bar{a}$ and $\bar{b}$ as two adjacent sides is $15$ square units,then the area (in square units) of the parallelogram,having $3 \bar{a} + 2 \bar{b}$ and $\bar{a} + 3 \bar{b}$ as two adjacent sides,is

Let the vectors $\overline{PQ}, \overline{QR}, \overline{RS}, \overline{ST}, \overline{TU},$ and $\overline{UP}$ represent the sides of a hexagon.
Statement-$1$: $\overline{PQ} \times (\overline{RS} + \overline{ST}) \neq \vec{0}$
Statement-$2$: $\overline{PQ} \times \overline{RS} = \vec{0}$ and $\overline{PQ} \times \overline{ST} = \vec{0}$

If $A(3,2,-1), B(-2,2,-3)$ and $D(-2,5,-4)$ are the vertices of a parallelogram,then the area of the parallelogram is

The unit vector perpendicular to both the vectors $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$ is

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\bar{a}|=1$,$|\bar{b}|=4$,and $\bar{a} \cdot \bar{b}=2$. If $\bar{c}=(2 \bar{a} \times \bar{b})-3 \bar{b}$,then the angle between $\bar{b}$ and $\bar{c}$ is