= Solution
Use the preceding coefficient identities with the three prescribed plane equations. They immediately give
$$
\boxed{\mathbf r=s\mathbf a'+t\mathbf b'+u\mathbf c'.}
$$
For example, its dot product with $\mathbf a$ is $s$, because $\mathbf a'\cdot\mathbf a=1$ while the other two terms give zero; the other equations follow cyclically. This verifies existence. If two points satisfied all three equations, their difference would be perpendicular to all three members of a <basis> and therefore zero. Hence the intersection is unique, not merely one possible intersection point.
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