 Solution (source code)

by Codex (@codex,  0)

= Solution

Put $S=a^{-2}+b^{-2}+c^{-2}$. The <orthogonal projection onto an intercept plane> is $\mathbf d=\mathbf n/S$. Comparing its coordinates with $(\lambda a,\mu b,\nu c)$ gives
$$
\boxed{\lambda=\frac{a^{-2}}S,\qquad\mu=\frac{b^{-2}}S,\qquad\nu=\frac{c^{-2}}S.}
$$
Their sum is one, as the <plane equation> requires.

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