If the 3D vectors n and i are known and the value for μ is also known, is it possible to use this equation to solve for t?
I have tried for hours and the closest I've gotten is computing the RHS of the equation and expressed t in terms of its components (x,y,z), but when I try to equate the LHS and the RHS to find values for x, y and z through a system of linear equations, there are no real solutions.
Thank you, any contribution will be greatly appreciated.
It is important to realize that all vectors (rays) in this figure can be considered in the same plane. This is because of the conservation of momentum in physics. An incident ray in the plane will only get deflected in the plane but not in or out of it becasue there is no external disturbance in that direction to do that. This simplify things considerably. Since you can take the plane to be perpendicular to the $z$ axis and the normal to be along the $x$ axis, and write your vectors: $$ n=(1,0,0)\\ i=(\cos\theta_1,\sin\theta_1,0)\\ t= (\cos\theta_2 ,\sin\theta_2,0) $$ Now you can take the cross product to find: $$n\times i=(0,0,\sin\theta_1)\\ n\times t=(0,0,\sin\theta_2) $$ and from the equation you have it follows that: $$ n_1\sin\theta_1=n_2\sin\theta_2 $$ which is Snell's law. Finding $t$ us just calculating $ \theta_2$ from the equation, given $\theta_1$, and use it in $t= (\cos\theta_2 ,\sin\theta_2,0)$.