You can specify the real and the imaginary parts of the two complex numbers z1=x1 + iy1 and z2=x2 + iy2 by moving the yellow and blue points on the left, resp. The two points z1 and z2 initially are shown in two different coordinate systems, namely at O and O'. To both points there is an associated triangle including the origin z=0 and the point z=1, namely (O1z1) and (O'1'z2).
We now can realize a scaled rotation of the blue triangle around the origin O. For this, a copy (QRz) of the blue triangle can be moved, such that (QR) sits at (Oz1): (1) First, shift the blue triangle such that Q lies as O, then the third points of the two triangles represent the complex numbers in the same coordinate system. (2) Now, rotate the blue triangle around O, such that the red line segment falls onto the line (O1). This is a rotation by the angle phi_1= (z1,O,1). (3) Finally, the scaling of the triangle can be realized by moving the point P0 on the left such that the point z of the blue triangle lies above z1. This is a scaling of the blue triangle by the length |z1|. If these operations are successful, the rotated and scaled point represents the product z= x + i y = z1z2 = (x1 x2 -y1 y2) + i (x1 y2 + x2 y1) of the two given complex numbers.