Let $\vec{\alpha}=3\hat{i}+\hat{j}$ and $\vec{\beta}=2\hat{i}-\hat{j}+3\hat{k}$. If $\vec{\beta}=\vec{\beta}{1}-\vec{\beta}{2}$, where $\vec{\beta}{1}$ is parallel to $\vec{\alpha}$ and $\vec{\beta}{2}$ is perpendicular to $\vec{\alpha}$, then $\vec{\beta}{1}\times\vec{\beta}{2}$ is equal to: