Uniform plane wave parallel/perpendicular phasor problem

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Homework Statement


img.png


Homework Equations


lambda= 2*pi / k
Ei(z), Ei_perpendicular (z), Ei_parallel(z) eq.
epsilon_r1=1, epsilon_r2=2.25
eta_1=120*pi, eta_2=80*pi


The Attempt at a Solution


I having trouble understanding some parts of this problem. i get k1=3.61, k2=5.42 and for d) theta_i=tan-1(2/3)=33.7deg. the only way I could try to find the z-component Ez by using the parallel component of Ei(z), so it would be 9*x - Ez*z * e^-j(k1*sin(theta_i)*x + k1*cos(theta_i)*z) and I tried to draw a picture of this on xyz plane and I got tan(90-theta_i)=Ex/Ez and solving I get Ez (amplitude)=6, but not sure if I did that right.

My other problem with the equations for parallel/perpendicular Ei(z) field, the amplitude for perpendicular I get on y-plane or -4*y, for the amplitude on parallel the equation given as (x*cos(theta_i) - z*sin(theta_i)) * E(parallel) so would it be x*9*cos(theta_i) - z*6*sin(theta_i) for the parallel Ei field or am I understanding it wrong?
 
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