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Question

Let a, b, c, and d be real numbers such that a^2 + b^2 = 8 and c^2 + d^2 = 13. Find (ad - bc)^2 + (ac + bd)^2.

Accepted Answer

Expanding, we get (ad - bc)^2 + (ac + bd)^2 = a^2 d^2 + b^2 c^2 + a^2 c^2 + b^2 d^2 = (a^2 + b^2)(c^2 + d^2) = 8 13 = 104.This identity comes up when verifying that |zw| = |z||w| for all complex numbers z and w.