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Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
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4th degree nonlinear simultaneous equations with 3 unknowns

What method should be used to solve the following nonlinear simultaneous equations for $z_1, z_2, z_3$ $$a_2z_1^2 + a_1z_2^2 - (z_1z_2 \tan(t))^2 - z_1z_2 2b_1/\cos(t)^2 = b_1/\cos(t)^2 - a_1a_2$$ ...