 Solution (source code)

by Codex (@codex,  0)

= Solution

For example, choose
$$
\boxed{\mathbf u=(-2,1,0)^T,\qquad \mathbf v=(3,0,1)^T.}
$$
Both <vectors> satisfy the <plane equation>. They are <linearly independent vectors>: in $\alpha\mathbf u+\beta\mathbf v=\mathbf0$, the second coordinate gives $\alpha=0$ and the third gives $\beta=0$.

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