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Benedict Irwin added Sketchy Generation.tex
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\section{Generating Operations}
In some sense, the determinant is able to generate the features if various algebras we know and love, For example vectors and their relationships, we have \begin{equation}
\textbf{r}_1 \times \textbf{r}_2 = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix} \\ \\
\textbf{r}_1 \cdot (\textbf{r}_2 \times \textbf{r}_3)=\begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix}
\end{equation}
Where the second, scalar triple product is the volume of the paralelipiped made by the three vectors. These were taken from (Leach, Molecular Modelling 2nd Edition) although it is widly known. Generally these are just shorthands or mnenomics. The mathematical value is limited.
The difference of these two then appears to be the introduction of rows of $i,j,k$ in the determinant and the removal of vector inputs. Above are a binary and ternary operation, experimenting with this concept in the same notation we can generate a unary operation and a 0-ary operation, i.e just an object. We have \begin{equation}
\textbf{r}_1 \to \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_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} \\ \textbf{i} & \textbf{j} & \textbf{k} \end{vmatrix}= \epsilon_{ijk}
\end{equation}
