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5.8: Angles of Reflection and Refraction
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/05%3A_Wave_Reflection_and_Transmission/5.08%3A_Angles_of_Reflection_and_Refraction
\[\begin{align} \beta_1\left(\hat{\bf x}\sin\psi^i + \hat{\bf z}\cos\psi^i\right)\cdot\left(\hat{\bf x}x + \hat{\bf y}y\right) \nonumber \\ =\beta_1\left(\hat{\bf x}\sin\psi^r - \hat{\bf z}\cos\psi^r\... $\begin{align} \beta_1\left(\hat{\bf x}\sin\psi^i + \hat{\bf z}\cos\psi^i\right)\cdot\left(\hat{\bf x}x + \hat{\bf y}y\right) \nonumber \\ =\beta_1\left(\hat{\bf x}\sin\psi^r - \hat{\bf z}\cos\psi^r\right)\cdot\left(\hat{\bf x}x + \hat{\bf y}y\right) \nonumber \\ =\beta_2\left(\hat{\bf x}\sin\psi^t + \hat{\bf z}\cos\psi^t\right)\cdot\left(\hat{\bf x}x + \hat{\bf y}y\right) \end{align} \nonumber$

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