If $\alpha=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}, \beta=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$, then express $\beta$ in the form $\beta=\beta_1+\beta_2$ where $\beta_1$ is parallel to $\alpha$ and $\beta_2$ is perpendicular to $\alpha$, then $\beta_1$ is given by