How can we express Snell's law as $\mu_1\sin i = \mu_2\sin r = constant$ and also as $\mu_1(\hat{IR}\times\hat{N}) = \mu_2(\hat{RR}\times\hat{N} )$ $\hat{IR}$ refers to the unit vector denoting the incident ray unit vector
$\hat{RR}$ refers to the unit vector denoting the refracted ray unit vector
$\hat{N}$ refers to the unit vector denoting the normal unit vector
Things that I know about cross product: $\overrightarrow{A} \times \overrightarrow{B} = AB\sin\theta$.
Snell's law in vector form 
I referred to this link but I can't understand any of the explanations about how we can arrive to the cross product form of Snell's Law
 A: 
Notation in Figure-01 :
\begin{align}
   n_1,n_2 & \boldsymbol{=} \texttt{indices of refraction}
   \nonumber\\
   \theta_1,\theta_2 & \boldsymbol{=} \texttt{angles of incidence and transmission}
   \nonumber\\
   \mathbf i & \boldsymbol{=} \texttt{unit vector on incident ray}
   \nonumber\\
   \mathbf t & \boldsymbol{=} \texttt{unit vector on transmitted ray}
   \nonumber\\
   \mathbf n & \boldsymbol{=} \texttt{unit vector normal to the interface of the two media} 
   \nonumber\\
   \mathbf k & \boldsymbol{=} \texttt{unit vector normal to the plane of }\mathbf i,\mathbf t,\mathbf n
   \nonumber 
\end{align}
Snell's law is expressed as
\begin{equation}
   n_1\sin\theta_1\boldsymbol{=}n_2\sin\theta_2
   \tag{01}\label{01}
\end{equation}
or
\begin{equation}
   \sin\theta_2\boldsymbol{=}\mu\sin\theta_1\,,\qquad \mu\boldsymbol{=}\dfrac{n_1}{n_2}
   \tag{02}\label{02}
\end{equation}
so
\begin{equation}
   \underbrace{\left(\sin\theta_2\right)\mathbf k}_{\left(\mathbf n\boldsymbol{\times}\mathbf t\right)}\boldsymbol{=}\mu\underbrace{\left[\left(\sin\theta_1\right)\mathbf k\right]}_{\left(\mathbf n\boldsymbol{\times}\mathbf i\right)}
   \tag{03}\label{03}
\end{equation}
that is Snell's law in vector form
\begin{equation}
\left(\mathbf n\boldsymbol{\times}\mathbf t\right)\boldsymbol{=}\mu\left(\mathbf n\boldsymbol{\times}\mathbf i\right)
  \tag{04}\label{04}
\end{equation}
