deletions | additions       diff --git a/Sketchy Generation.tex b/Sketchy Generation.tex  index 7f86176..10d81a1 100644  --- a/Sketchy Generation.tex  +++ b/Sketchy Generation.tex 
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Without this permutation, which could be expressed as a permutation matrix $P$ multiplied onto the result of the $S$ operator we can ask what is the value of $\hat{S}(\textbf{r}_1)\hat{S}(\textbf{r}_2)$? \begin{equation}  \hat{S}(\textbf{r}_1)\hat{S}(\textbf{r}_2)=\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ \textbf{i} & \textbf{j} & \textbf{k} \\ x_1 & y_1 & z_1 \end{vmatrix} = \hat{S}(\textbf{r}_1)\hat{S}(\textbf{r}_2)=  \begin{bmatrix} 0 & z_1 & -y_1 \\ -z_1 & 0 & x_1 \\ y_1 & -x_1 & 0 \end{bmatrix}\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ \textbf{i} & \textbf{j} & \textbf{k} \\ x_1 & y_1 & z_1 \end{vmatrix} =  \begin{bmatrix} 0 & z_2 & -y_2 \\ -z_2 & 0 & x_2 \\ y_2 & -x_2 & 0 \end{bmatrix} =  \end{equation}