Multiplying Complex Numbers

This graph shows how we can interpret the multiplication of complex numbers geometrically. Given two complex numbers: , where Consider their product
1Dilate  by a scale factor of
2Rotate by about
3Dilate by a scale factor of
4Translate by
Focus on the two right triangles in the diagram:
  1. The right triangle formed by , and the positive real axis.
  2. The right triangle formed by , and
The first right triangle has sides of length: , , . The second right triangle has sides of length , , and . Since we have the proportion: , we can conclude the triangles are similar since two pairs of corresponding sides are proportional and their included angles (the right angles) are congruent. This has two implications:
  1. The ratio of similitude is , which means that (this is an alternative to the algebraic proof you did for homework)
  2. The angle formed by , and is congruent to , since they are corresponding angles of similar triangles
This leads us to our other conclusion, that Key results: