Continuing with the hint No 2, it is useful to notice that if z in the expression \varepsilon = \mathrm e^{\mathrm i \varphi} + z corresponds to \vec l \times \vec v_2 then \vec l \times \vec v_1 is obtained by making z two times bigger and rotating by 90 degrees which is the same as multiplying by the imaginary unit, hence what we need is 2z\mathrm i. From here, the full solution is just a few lines away.
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