Let a = i +2 j -3 k and b =2 i -3 j +5 k. If r × a = b × r, r ·(α i +2 j + k )=3 and r ·(2 i +5 j -α k )=-1, α ∈ R, then the value of α+| r |^2 is…

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