Explicit solution of vector

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Suppose there are vectors $p_i,p_j,z$ and scalar $r$.

We have such following relation ship: $$ z^Tp_i - z^T p_j = r^2 $$

How to solve the explicit solution of vector $z$?

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The black vector is $(p_i - p_j)$ and the plane is at a distance $r^2$ from the base of that vector. The blue vectors are just a few of the infinite set of vectors with the same dot product with $(p_i - p_j)$.

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