So, for a given ω, if we redefine k as the magnitude of the wave vector in the reference medium (for which n = ‍ 1), then the wave vector has magnitude n 1 k in the first medium (region ‍ y < 0 ‍ in the diagram) and magnitude n 2 k in the second medium. From the magnitudes and the geometry, we find that the wave vectors are k i = n 1 k ( i sin ⁡ ⁡ θ θ i + j cos ⁡ ⁡ θ θ i ) k r = n 1 k ( i sin ⁡ ⁡ θ θ i − − j cos ⁡ ⁡ θ θ i ) k t = n 2 k ( i sin ⁡ ⁡ θ θ t + j cos ⁡ ⁡ θ θ t ) = k ( i n 1 sin ⁡ ⁡ θ θ i + j n 2 cos ⁡ ⁡ θ θ t ) , where the last step uses Snell's law. The corresponding dot products in the phasor form (3) are