Question

Verify the following geometric formula for complex multiplication. Let z,w∈C. Form a triangle in the complex plane with vertices 0,1 and z. Then zw is the third vertex of a similar triangle with other vertices 0 and w, which is oriented in the same way and so that the vector w corresponds to the vector 1 .

Answer 1

The formula for complex multiplication in the complex plane can be verified by forming a triangle with vertices 0, 1, and z, and then finding the third vertex of a similar triangle with vertices 0 and w. The ratio of the adjacent side to the hypotenuse of the first triangle is the cosine of the angle of interest. Formulas are provided to calculate the values of L and Lz.

Step-by-step explanation:

The formula being discussed is related to complex multiplication in the complex plane. To verify the formula, we need to form a triangle with vertices 0, 1, and z. The third vertex of a similar triangle with vertices 0 and w, which is oriented in the same way, will be zw. The ratio of the adjacent side (Lz) to the hypotenuse (L) of the first triangle is the cosine of the angle of interest. We can find L and Lz using the given formulas.