Complex algebra

[Batzo]

Hi
Im trying to understand how this is correct with scalar multiplication.
Where: a, b are vectors in xy plane and a, b are complex numbers
a=(x1,y1)
a*=x-iy

ab=0.5(ab*+ba*)

 

[HallsofIvy]

How are vector a and b related to complex numbers a and b?
Do you mean that a= (x1, y1), b= (x2, y2) while a= x1+iy1 and b=x2+ iy2?

If so then ab*+ ba*= (x1+ iy1)(x2- iy2)+ (x2+ iy2)(x1- iy1)= x1x2- ix1y2+ix2y1+ y1y2+ x1x2- ix2y1+ ix1y2+ y1y2= (2x1x2+2y1y2)+ i(-x1y2+x2y1+x1y2- x2y1)= 2x1x2+ 2y1y2.

So (1/2)(ab*+ ba*)=x1x2+ y1y2.

Now, what is ab=(x1, y1).(x2, y2)?

 

[Batzo]

I didnt remember that with dot multiplication perpendicular vectors equal zero:
i*j=0, so
ab=(x1, y1).(x2, y2) =x1x2+y1y2+(x1y2+x2y1) =x1x2+y1y2