Let \(\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}\).If \(\vec{r} \times \vec{a}=\v…

Let \(\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}\).If \(\vec{r} \times \vec{a}=\vec{b} \times \vec{r}, \vec{r} \cdot(\alpha \hat{i}+2 \hat{j}+\hat{k})=3\) and \(\vec{r} \cdot(2 \hat{i}+5 \hat{j}-\alpha \hat{k})\), \(=-1, \alpha \in R\) then the value of \(\alpha+|\vec{r}|^2\) is equal to:

[JEE Main 2021, 16 Mar (Shift 2)]