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Plane through 3 Points

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A plane in $ \mathbb{R}^3$ through three points $ P_1=(x_1,y_1,z_1)$, $ P_2=(x_2,y_2,z_2)$ and $ P_3=(x_3,y_3,z_3)$ can be expressed in implicit form as

$\displaystyle E:\left\vert\begin{array}{cccc} x&y&z&1\\ x_1 &y_1&z_1&1\\ x_2&y_2&z_2&1\\ x_3&y_3&z_3&1\end{array}\right\vert=0\,.$

Expanding along the first row we obtain the representation

$\displaystyle E: ax+by+cz=d$

with

$\displaystyle a=\left\vert\begin{array}{ccc} y_1&z_1&1\\ y_2&z_2&1\\ y_3&z_3&1... ...egin{array}{ccc} x_1&y_1&z_1\\ x_2&y_2&z_2\\ x_3&y_3&z_3\end{array}\right\vert.$

(Authors: Burkhardt/Höllig/Streit)
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  automatisch erstellt am 27.  1. 2005