Vector Algebra: JEE Main 2026 Mathematics question

Let $\overrightarrow{\mathrm{AB}}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{i}+2\hat{j}+\lambda\hat{k}$, $\lambda\in\mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on the diagonal $\overrightarrow{\mathrm{AC}}$ of the parallelogram ABCD be of length one unit. If $\alpha, \beta$, where $\alpha>\beta$, be the roots of the equation $\lambda^2x^2-6\lambda x+5=0$, then $2\alpha-\beta$ is equal to