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$\hat{IR}$ 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

Buddy
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  • This is exactly (a subset of) the question you link to. If you don't say what you don't understand about the answers there, its not going to be possible to give you an answer. – jacob1729 Jun 04 '21 at 14:52
  • Regarding the first part of the title question: I am confused what there is to know, Snell's law is the statement that $\mu_1 \sin i = \mu_2 \sin r$. – jacob1729 Jun 04 '21 at 14:53
  • I found the explanations too difficult to understand, so i was hoping that if someone could simplify it for me... – Buddy Jun 05 '21 at 07:26
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    Welcome to PSE. Note that the cross product of two vectors is a vector too, not a scalar as it seems that you wrongly know. More precisely : $\mathbf a \boldsymbol{\times}\mathbf b \boldsymbol{=} \left(|\mathbf a ||\mathbf b|\sin\theta\right)\mathbf n$ where $\mathbf n$ the unit vector normal to the plane of $\mathbf a ,\mathbf b$ obeying the right hand rule. – Frobenius Jun 05 '21 at 21:51
  • Related :Why one should follow Snell's law for shortest time?. – Frobenius Jun 06 '21 at 05:32

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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}

Frobenius
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