deletions | additions       diff --git a/4-Dimensions.tex b/4-Dimensions.tex  index e6541e6..7a8ad65 100644  --- a/4-Dimensions.tex  +++ b/4-Dimensions.tex 
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\end{vmatrix}  \end{equation}  Therefore for a set of 3 vectors 3-vectors  \begin{equation} _tr_1=(x_1,y_1,z_1) _tr_1=x_1\textbf{j}+y_1\textbf{k}+z_1\textbf{l}  \\ _xr_1=(t_1,y_1,z_1) _xr_1=t_1\textbf{i}+y_1\textbf{k}+z_1\textbf{l}  \\ _yr_1=(t_1,x_1,z_1) _yr_1=t_1\textbf{i}+x_1\textbf{j}+z_1\textbf{l}  \\ _zr_1=(t_1,x_1,y_1) _zr_1=t_1\textbf{i}+x_1\textbf{j}+y_1\textbf{k}  \\ \end{equation} 
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\end{equation}  Which can obviously take on other forms by going along the other rows to obtain the triple products according to $134$,$124$ or $123$ sets of vectors, with appropriate signs. For the binary transform we have \begin{equation}  \begin{vmatrix}   \textbf{i} & \textbf{j} & \textbf{k} & \textbf{l} \\  \textbf{i} & \textbf{j} & \textbf{k} & \textbf{l} \\  t_1 & x_1 & y_1 & z_1 \\  t_2 & x_2 & y_2 & z_2   \end{vmatrix}  =\textbf{i}({_t\textbf{r}_1} \times {_t\textbf{r}_2} )  -\textbf{j}({_x\textbf{r}_1} \times {_x\textbf{r}_2} )  +\textbf{k}({_y\textbf{r}_1} \times {_y\textbf{r}_2} )  -\textbf{l}({_z\textbf{r}_1} \times {_z\textbf{r}_2} )  \end{equation}