Question #66165

Let $M = \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{pmatrix}$ be a matrix whose rows are the vectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$. Let $\vec{p}_i$ be the orthogonal projection of $\vec{v}_i$ onto the plane $x+y+z=0$. The determinant $\Delta$ of the matrix with rows $\vec{p}_1, \vec{p}_2, \vec{p}_3$ is identically zero, providing no unique solution for $a, b, c$. However, it is given that $\vec{p}_1 + \vec{p}_2 + \vec{p}_3 = \vec{0}$. Let $L_1$ be the line $\vec{r} = \lambda \vec{v}_1$ and $L_2$ be the line $\vec{r} = \hat{k} + \mu \vec{v}_2$. If the shortest distance between $L_1$ and $L_2$ is $\frac{1}{\sqrt{5}}$, find the value of $b^2$.