Determining a plane by a point and two vectors with different directions.
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Idea: A plane is determined uniquely if we know a
point in the plane and two vectors in the plane which have different
directions.
Definition 1. Every plane in \(\mathbb{R}^3\) can be described in the
following form: \[E=\left\{\mathbf{p}+\lambda
\mathbf{a}+\mu \mathbf{b}:\ \lambda,\mu\in\mathbb{R}\right\} =
\mathbf{p} + \text{Span}(\mathbf{a},\mathbf{b}),\] where the
points of \(E\) are represented by
their position vectors. This representation is called parameter
form.
Example 2. In \(\mathbb{R}^3\) there exists a unique plane
\(E\) which contains the points \(R:=(1,1,1)\), \(S:=(1,3,2)\) and \(T:=(-1,4,3)\). We determine a parameter
form of \(E\).
point in the plane: \(\mathbf{p}=\begin{pmatrix} 1\\1\\1
\end{pmatrix}\)
vectors in the plane: \(\mathbf{a}=
\begin{pmatrix} 0\\2\\1 \end{pmatrix}\) and \(\mathbf{b}= \begin{pmatrix} -2\\3\\2
\end{pmatrix}\)
A parameter form of \(E\) is: \[\begin{aligned}
E & = \left\{\begin{pmatrix} 1\\1\\1 \end{pmatrix} + \lambda
\begin{pmatrix} 0\\2\\1 \end{pmatrix}
+ \mu \begin{pmatrix} -2\\3\\2 \end{pmatrix}:\
\lambda,\mu\in\mathbb{R} \right\} \\
& = \begin{pmatrix} 1\\1\\1 \end{pmatrix} +
\operatorname{Span}\left( \begin{pmatrix} 0\\2\\1 \end{pmatrix},
\begin{pmatrix} -2\\3\\2 \end{pmatrix} \right)~ .
\end{aligned}\]
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