Let (α, β, γ) be the co-ordinates of the foot of the perpendicular drawn from the point (5,4,2) on the line r =(- i +3 j + k )+λ(2 i +3 j - k ). Then…

Let $\displaystyle (\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $\displaystyle (5,4,2)$ on the line $\displaystyle \vec{r}=(-\hat{i}+3 \hat{j}+\hat{k})+\lambda(2 \hat{i}+3 \hat{j}-\hat{k})$.Then the length of the projection of the vector $\displaystyle \alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ on the vector $\displaystyle 6 \hat{i}+2 \hat{j}+3 \hat{k}$ is: