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

Let $\vec{b}=3 \hat{i}-2 \hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be two vectors. If $\vec{a}$ is a vector such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$,then $|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|=$

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

Suppose $L_1$ and $L_2$ are two lines having the direction ratios $1, -2, -2$ and $0, 2, 1$ respectively. If the direction cosines of a line perpendicular to both $L_1$ and $L_2$ are $l, m, n$, then $|l| + |m| + |n| =$

Let $ABCD$ be a quadrilateral with $AB = a$, $AD = b$ and $AC = 3a + 2b$. If its area is $\alpha$ times the area of the parallelogram with $AB$ and $AD$ as adjacent sides, then the value of $\alpha$ is equal to

If $|\bar{u}| = 8$ and $|\bar{v}| = 12$ with an angle of $150^{\circ}$ between them,then find $|\bar{u} \times \bar{v}|$.