Let AB =2 i +4 j -5 k and AD = i +2 j +λ k, λ ∈ R. Let the projection of the vector v = i + j + k on the diagonal AC of the parallelogram ABCD be of…

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