General

@aznlemony274:
It's not the same as a dot procuth because you cannot use the same rules for the inner product. (e.g. a dot product can give you a non-real length on a vector ([b]w[/b]=(1,2i)))

Here are 2 pages from my book, introducing the inner product:
http://dl.dropbox.com/u/41504745/sf1629/Photo%202011-11-09%2016%2048%2023.jpg
http://dl.dropbox.com/u/41504745/sf1629/Photo%202011-11-09%2016%2048%2050.jpg

Also, I'd do something like this:
i||u-iv||^2=i<u-iv,u-iv>=i(||u||^2+<u,-iv>+<-iv,u>+||iv||^2)

Quaternions <3

@aznlemony274:
Inner product =/= scalar product, and what did you do in the last 2 steps?
[i]i||u+v||^2= _____________ i||u||^2 -2 u v -i||v|| ^2[/i]
[i]-i||u-v||^2= _____________-i ||u||^2+2u v +i||v|| ^2 [/i]

It's supposed to be i||u-iv||^2 and -i||u-iv||^2.