\( \textbf{(1)} \left| \left| \vec{a} \times \vec{b} \right| \right|^2 = (a_2 b_3 - a_3 b_2 )^2 + (a_3 b_1 - a_1 b_3)^2 + (a_1 b_2 - a_2 b_1)^2 \)

\( \textbf{(2)} =a_{2}^{2} b_{3}^{2} - 2a_2 a_3 b_2 b_3 + a_{3}^{2} b_{2}^{2} + a_{3}^{2} b_{1}^{2} - 2a_1 a_3 b_1 b_3 + a_{1}^{2} b_{3}^2 + a_{1}^{2}b_{2}^{2} - 2a_1 a_2 b_1 b_2 + a_{2}^{2} b_{1}^{2} \)

\( \textbf{(3)} =(a_{1}^2 + a_{2}^2 + a_{3}^2 ) (b_{1}^2 + b_{2}^2 + b_{3}^2 ) - (a_1 b_1 + a_2 b_2 + a_3 b_3)^2 \)

And so on.

How do we go from step (2) to step (3), algebraically?

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