\[n\]
points \[P_1. \; P_2, \; ..., \; P_{n+1}\]
points in \[\mathbb{R}^n\]
which do not lie in an \[n-1\]
flat, the equation for the hyperplane though the points is\[det \left( \begin{array}{ccccc} \mathbf{P}_1 & \mathbf{P}2 & \ldots & \mathbf{P}_{n} & \mathbf{X} \\ 1 & 1 & \ldots & 1 & 1 \end{array} \right) = 0\]
where each
\[P\]
is written as the column vector \[\mathbf{OP}\]
and \[\mathbf{X}= \begin{pmatrix}x_1\\x_2\\ \vdots \\ x_n\end{pmatrix}\]
/The equation of a hyperplane in
\[\mathbb{R}^2\]
passing through the points \[(0,1), \; (2,3)\]
is\[det \left( \begin{array}{ccc} 0 & 2 & x_1 \\ 1 & 3 & x_2 \\ 1 & 1 & 1 \end{array} \right)=-2x_1-2x_2-2=0.\]