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Question:** A science fiction writer describes a planet where two force fields, modeled by complex numbers \(z_1 = 3 + 4i\) and \(z_2 = 1 - 2i\), interact. Compute the product of these two force fields.
To find the product of the two complex numbers \(z_1\) and \(z_2\), we use the distributive property of multiplication for complex numbers:
z_1 \cdot z_2 = (3 + 4i)(1 - 2i)
Expanding this, we get:
= 3 \cdot 1 + 3 \cdot (-2i) + 4i \cdot 1 + 4i \cdot (-2i)
= 3 - 6i + 4i - 8i^2
Recall that \(i^2 = -1\). Substituting, we have:
= 3 - 6i + 4i + 8
= (3 + 8) + (-6i + 4i)
= 11 - 2i