Example 1.2.2: Multiplication in 2D and C 
Explore the differences between complex multiplication in C,
the dotproduct in R^{2}, and the crossproduct in
R^{2} as embedded in R^{3}.
Context

We now have four types of "multiplication" for vectors (x,y) in
R^{2}.
Dot Product
For 'standard' R^{2} vectors we have the (hopefully)
familiar dot product:
(x,y) (u,v) = xu + yv
This type of multiplication results in a number (scalar), not a vector. It is
therefore not useful for our purpose, since we need multiplication to result in
another vector.
Geometrically the dot product gives the (cos of the) angle between
(x,y) and (u,v). Useful indeed, but not for our
current purpose.
Cross Product
If we think of 2D vectors (x,y) as 3D vectors with
0 as third component (x,y,0) we have the cross
product:
(x,y,z) (u,v,w) = (ywzv, xw+zu, xvyu)
This type of product does yield a vector alright. But if we consider vectors
whose third components are zero, such as (x,y,0) and
(u,v,0) we get:
(x,y,0) (u,v,0) = (00, 0+0, xvyu) = (0,0,xvyu)
i.e. while the first two vectors live in the xyplane, the result
of a cross product lives perpendicular to that plane! Again, the result of our
crossproduct is not in our embedded R^{2} (third component
is not zero), so this product is no good (for our current purposes), too.
Silly Product
We already discussed the silly multiplication:
(x,y) (u,v) = (xu, yv)
which would violate the field axioms, so  definitely no good.
Complex Product
The original definition of complex multiplication
(x,y) (u,v) = (xu  yv, xv + yu)
will be the one that turns the usual 2dimensional plane into the complex
plane. We will discuss the geometric interpretation of this type of
multiplication soon.
Interactive Complex Analysis, ver. 1.0.0
(c) 20062007, Bert G. Wachsmuth
Page last modified: May 29, 2007