ComplexInnerProductVsDotAndCrossProduct.tex

%
% Copyright © 2016 Peeter Joot.  All Rights Reserved.
% Licenced as described in the file LICENSE under the root directory of this GIT repository.
%

\makeproblem{Complex inner product vs. dot and cross product.}{problem:introGAproblems:ComplexInnerProductVsDotAndCrossProduct}{
Given two 2D vectors \( (a,b) \) and \( (a', b') \), and a complex number representation of these vectors \( z = a + ib, w = a' + i b' \), show that the components of the complex inner product have the representation
given by \cref{eqn:GAmotivation:220}.
} % problem

\makeanswer{problem:introGAproblems:ComplexInnerProductVsDotAndCrossProduct}{
\begin{dmath}\label{eqn:introGAproblems:20}
z w^\conj
=
(a + ib)(a' - ib')
=
a a' + b b'
+ i \lr{ a' b - a b' }
\leftrightarrow
( a a' + b b', a' b - a b' ).
\end{dmath}
} % answer