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.

- Why is
as shown? Why is as shown? Why is as shown? - Look for similar triangles. Why are they similar? What is the scale factor?
- How does the angle of the product
relate to the angles of and of ? Explain. - What about their magnitudes? Explain.