Consider the complex numbers in the following picture, as well as the line segments connecting them to the origin.
Find the number of the quadrant each of these pairwise products is in : \(z_1z_2, \, z_1 z_3, \, z_1 z_4, \, z_2 z_3, \, z_2 z_4, \, z_3 z_4.\)
+6248 \(w_1,w_2 \in \mathbb{C} \Rightarrow w_1w_2 = |w_1||w_2|e^{\arg(w_1)+\arg(w_2)}\\ \text{So in determining what quadrant the result is in we add the individual angles}\)
\(\arg(z_1) \approx \dfrac \pi 6\\ \arg(z_2) \approx \dfrac \pi 4\\ \arg(z_3) \approx \dfrac{7\pi}{12}\\ \arg(z_4) \approx \dfrac{7\pi}{6}\)
\(\text{Now adding these according to the products you're given}\\ \begin{array}{ccccc} \text{Product}&\text{angle 1}&\text{angle 2}&\text{sum}&\text{quadrant}\\ z_1z_2 &\dfrac\pi 6&\dfrac \pi 4 &\dfrac{5\pi}{12} &1\\ z_1z_3 &\dfrac\pi 6&\dfrac{7\pi}{12} &\dfrac{9\pi}{12} &2\\ z_1z_4 &\dfrac\pi 6&\dfrac{7 \pi}{6} &\dfrac{4\pi}{3} &3\\ z_2z_3 &\dfrac\pi 4&\dfrac {7\pi}{12} &\dfrac{5\pi}{6} &2\\ z_2z_4 &\dfrac\pi 4&\dfrac {7\pi}{6} &\dfrac{17\pi}{12} &3\\ z_3z_4 &\dfrac{7\pi}{12}&\dfrac {7\pi}{6} &\dfrac{7\pi}{4} &4\\ \end{array}\)
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