I want to take the inverse of a dot product, where both vectors have complex components. In other words, if $\textbf{A} \cdot \textbf{B} = d$, and I know $\textbf{A}$ and $d$, I want to find a $\textbf{B}$. I know that I cannot do so uniquely, which is fine; I have a procedure for creating a set of vectors that will satisfy $\textbf{A} \cdot \textbf{B} = d$. But, it relies on finding vectors that are orthogonal to $\textbf{A}$ and each other. Normally this would not be a problem; just take $\textbf{A}$, zero all but two of its components, switch the last two, and negate one of them. Make sure it's a different pair every time. For more orthogonal vectors, you can take cross products.
The difficulties arise when I consider vectors with complex components. I want to normalize each of the orthogonal vectors, which means divide by their magnitudes. I have read that you need to divide by $\sqrt{|v_x|^2 + |v_y|^2 + |v_z|^2 ...}$. What is the justification for this? Since I am just normalizing vectors, do I absolutely have to take the magnitudes of the components? Also, My technique of finding the inverse dot product relies on the identity $\textbf{A} \cdot \textbf{B} = d = |\textbf{A}||\textbf{B}|$. What modifications might I need to make? I can post more details if people want. Also, if anyone has links to stuff to read about, especially with regards to the reasons why, I would be most appreciative.