If $\alpha$ is the angle between two vectors $p = 3\hat{i} + 4\hat{j} - \hat{k}$ and $q = 2\hat{i} - \hat{j} + \hat{k}$,then $\sin(\alpha) = $

If the vertices of $\Delta ABC$ are $A=(2,3,5), B=(-1,3,2), C=(3,5,-2)$,then the area of the $\Delta ABC$ (in sq. units) is

Let $\overline{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}$,$\overline{b}=3 \hat{i}-\beta \hat{j}+4 \hat{k}$ and $\overline{c}=\hat{i}+2 \hat{j}-2 \hat{k}$,where $\alpha, \beta \in R$,be three vectors. If the projection of $\overline{a}$ on $\overline{c}$ is $\frac{10}{3}$ and $\overline{b} \times \overline{c}=-6 \hat{i}+10 \hat{j}+7 \hat{k}$,then the value of $\alpha^2+\beta^2-\alpha \beta$ is equal to

Let $\overrightarrow{OA}=\overrightarrow{a}$,$\overrightarrow{OB}=12 \overrightarrow{a}+4 \overrightarrow{b}$,and $\overrightarrow{OC}=\overrightarrow{b}$,where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$,then the ratio of the area of the quadrilateral $OABC$ to the area of $S$ is:

For vectors $\bar{a}$ and $\bar{b}$,$|\bar{a}| = \frac{2}{3}$,$|\bar{b}| = 3$ and $|\bar{a} \times \bar{b}| = 1$,then the angle between $\bar{a}$ and $\bar{b}$ is . . . . . . .

$A$ non-zero vector $\vec{a}$ is parallel to the line of intersection of the planes defined by the vectors $\vec{i}, \vec{i} + \vec{j}$ and $\vec{i} - \vec{j}, \vec{i} + \vec{k}$. The angle between $\vec{a}$ and the vector $\vec{i} - 2\vec{j} + 2\vec{k}$ is .....