Linear equation from 2 points
Often you have given 2 points $A$ and $B$, from which you should set up a linear equation.
Therefore you have to determine the position vector $\vec{OA}$ (or $\vec{OB}$) and the vector between two points $\vec{AB}$ and insert them into the parametric equation:
$\text{g: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$
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Info
Parametric equation:
- One of the points is needed as a support vector (or its position vector as support vector).
- The vector between the two points corresponds to the direction vector of the line.
Example
Determine the linear equation $g$ using the points $A(1|1|0)$ and $B(10|9|7)$.
Position vector
$\vec{OA}=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$Vector between two points
$\vec{AB}$ $=\begin{pmatrix} 10-1 \\ 9-1 \\ 7-0 \end{pmatrix}$ $=\begin{pmatrix} 9 \\ 8 \\ 7 \end{pmatrix}$Insert
$\text{g: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$$\text{g: } \vec{x} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 9 \\ 8 \\ 7 \end{pmatrix}$