Determinental Expression for Hyperplane in Rn

Given  
\[n\]
  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
\[det \left( \begin{array}{ccc} 0 & 2 & x_1 \\ 1 & 3 & x_2 \\ 1 & 1 & 1 \end{array} \right)=-2x_1-2x_2-2=0.\]

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