If two sides of a triangle are represented by vectors $\hat{i}+2\hat{j}+2\hat{k}$ and $3\hat{i}-2\hat{j}+\hat{k}$,then find the area of the triangle.

If $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}\times \overrightarrow{d}$ and $\overrightarrow{a}\times \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{d}$,then prove that $\overrightarrow{a}-\overrightarrow{d}$ is parallel to $\overrightarrow{b}-\overrightarrow{c}$.

The four vertices of a tetrahedron are respectively $O(0,0,0)$, $A(1,2,1)$, $B(2,1,3)$ and $C(1,1,2)$.Find the volume of the tetrahedron.

Prove that:

$\left[\overrightarrow{a}+\overrightarrow{b} \overrightarrow{b}+\overrightarrow{c} \overrightarrow{c}+\overrightarrow{a} \right]=2\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]$

Prove that:

$\left[\overrightarrow{(a}\times \overrightarrow{b)} (\overrightarrow{b}\times \overrightarrow{c}) (\overrightarrow{c}\times \overrightarrow{a)} \right]={\left[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}\right]}^2$