Complex Multiplication

Explore the geometry of complex multiplication. Below, and are complex numbers, and , where and are real numbers. Build the product by considering the real and imaginary parts of separately.
  1. Why is as shown? Why is as shown? Why is as shown?
  2. Look for similar triangles. Why are they similar? What is the scale factor?
  3. How does the angle of the product relate to the angles of and of ? Explain.
  4. What about their magnitudes? Explain.