Does a complex number multiplication have a geometric representation and why? [duplicate]

In order to understand multiplication of complex numbers, it is better to represent complex numbers in "polar" form, namely $$ z=r(\cos(\theta)+i\sin(\theta)) $$ where $r$ is a positive real number representing the length of the vector defined by $z$ in the complex plane and $\theta$ the angle it forms with the real axis.

When you multiply $z$ with a similarly written $z^\prime$ and perform all the algebra you eventually get $$ zz^\prime=rr^\prime(\cos(\theta+\theta^\prime)+i\sin(\theta+\theta^\prime)) $$ The geometrical interpretation of multiplication is now very visible. E.g. if $|z^\prime|=1$ (i.e. $r^\prime=1$) multiplication by $z^\prime$ is just rotation by some angle.

Complex multiplication has no connection to the cross product of vectors.

You can understand complex multiplication as an addition in the logarithmic domain (the log of a complex number is the log of its module plus $i$ times the argument): you multiply the modules together and you add the arguments.

In geometric terms, one of the numbers applies a similarity transform to the other (scaling and rotation). Unfortunately, this interpretation is asymmetric. It is constructible with a ruler and a compass.

Take the green ($z_1$) and black ($1$) vectors and rotate them to align on the blue ($z_2$) one; this adds the arguments. Then by an homothety, you multiply the lengths and get the red ($z_1z_2$) vector.