If we have a ray striking a plane then the unit vector in direction of the incident ray can be thought of as rotating in a plane that contains the normal to interface and the part of the unit vector along the plane of the interface.

However, I can not understand why this unit vector is not rotated around lines parallel to the plane of the interface which it strikes. To summarize, why does normal, incident and refracted ray lie in the same plane? I am asking for a physical answer, not a mathematical one.

I have already seen a mathematical answer here in this post but I don't think it really explained anything of 'why' it happens and, even if it does, uses some strange operations which I have not heard of it.

I seek either a less mathematically sophisticated answer/ one that directly explains the reasoning for this physical phenomenon happens. Also, I already know of snell's law of refraction, I'm asking for the intuition for why it should be true.

An afterthought: If light travels as a wave then shouldn't It technically refract in a lot of directions when it hits a surface?