This is a practical problem related to complex vectors. Imagine you want to find the resultant electric field of multiple electromagnetic waves that have parallel and perpendicular components propagating different directions. I transform each vector pair into a common frame work and do the addition.
In summary, the electric field polarization vector of an electromagnetic wave can be written by its parallel ($E_{\parallel}$) and perpendicular component vectors ($E_{\perp}$). Both $E_{\parallel}$ and $E_{\perp}$ are complex vectors. As orientations of $E_{\parallel}$ and $E_{\perp}$ are known, directional cosines cosα1,cosβ1,cosγ1 and cosα2,cosβ2,cosγ2 can be calculated. Then we can use following transformation to calculate x-,y- and z- vector components.
$\left(\begin{array}{c} E_{x}\\E_{y}\\E_{z} \end{array} \right)=\left( \begin{array}{ccc} cos\alpha_1 & cos\alpha_2 \\ cos\beta_1 & cos\beta_2 \\ cos\gamma_1 & cos\gamma_2 \end{array} \right)\left( \begin{array}{c} E_{\parallel}\\ E_{\perp} \end{array}\right)$
Now, I want to find the parallel and perpendicular components of resultant electric field and its direction. What I only have is $E_x$, $E_y$, and $E_z$. I don't have any angle information.
Using numerical examples, I found the following relationship.
$\left(\begin{array}{c} E_{\parallel}\\E_{\perp} \end{array} \right)=\left( \begin{array}{cc} cos\alpha_1 & cos\beta_1 & cos\gamma_1 \\ cos\alpha_2 & cos\beta_2 & cos\gamma_2 \end{array} \right)\left( \begin{array}{c} E_x \\ E_y \\ E_z \end{array}\right)$
Now my problem is finding angles cosα1,cosβ1,cosγ1 and cosα2,cosβ2,cosγ2. If I know angles, I can calculate $E_{\parallel}$ and $E_{\perp}$. My question is how to calculate those angles?
Other information I have is given below.
I can calculate $|E_{total}|$ as
$ |E_{total}| = \sqrt{|E_x|^2+|E_y|^2+|E_z|^2)} $
Then directional cosines ($cos\alpha_0, cos\beta_0, cos\gamma_0$) as
$cos \alpha_0 = \frac{|E_{x}|}{|E_{total}|}$, $cos \beta_0 = \frac{|E_{y}|}{|E_{total}|}$, and $cos \gamma_0 = \frac{|E_{z}|}{|E_{total}|}$.
But these angles are with $|E_{total}|$.