If $|\vec{a}|=10, |\vec{b}|=2$ and $\vec{a} \cdot \vec{b}=12$,then $|\vec{a} \times \vec{b}|=$ . . . . . . .

If $\bar{a}=2\hat{i}+3\hat{j}-\hat{k}$,$\bar{b}=-\hat{i}+2\hat{j}-4\hat{k}$ and $\bar{c}=\hat{i}+\hat{j}+\hat{k}$,then $(\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=$

If $a = 2i + 2j - k$ and $b = 6i - 3j + 2k,$ then the value of $a \times b$ is

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2\hat{j}+3\hat{k}$. If the position vectors of the points $A, B$ and $C$ are $-2\hat{i}+\hat{j}-3\hat{k}$,$2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}+2\hat{j}-\hat{k}$ respectively,then the projection of the vector $\overline{OP}$ on a vector perpendicular to the vectors $\overline{AB}$ and $\overline{AC}$ is $......$.

Find a unit vector coplanar with $i + 2j + k$ and $i + j + 2k$ and perpendicular to $i + j + k$.

The diagonals of a parallelogram are $\vec{d_1} = \hat{j} + \hat{k}$ and $\vec{d_2} = \hat{i} + \hat{j}$. The area of the parallelogram is . . . . . . sq. units.