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When is $(z_1 - z_2)(z_3 - z_4)$ a negative real number? I know that for this number to be real, then its imaginary part must be 0; however, for the real part, I get

$$ax - aw - cx + cw + by - bz - dy + dz.$$

How do I know when this is negative? I need this in order to solve this problem.

Dylan C. Beck
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Massimo2015MX
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  • Can you explain where this expression comes from? I understand that you have expressed each of these complex numbers in the form $z = a + bi,$ but can you explicitly write down which one is which? I believe that there is an easier way to figure this out. – Dylan C. Beck May 28 '20 at 21:28
  • Treat them as complex vectors. You want their product to be $ r Cis (\pi )$. What does that tell you about their angles? – Calvin Lin May 28 '20 at 21:29
  • @Carlo $z_1 = a + bi$ $z_2 = c + di$ $z_3 = x + yi$ $z_4 = w + zi$ – Massimo2015MX May 28 '20 at 22:59
  • @CalvinLin that they suplementary – Massimo2015MX May 29 '20 at 00:11

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