Math Lines in three dimensions Setting up a linear equation

# 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 $B(10|9|7)$.

1. #### Position vector

$\vec{OA}=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$
2. #### Vector between two points

$\vec{AB}$ $=\begin{pmatrix} 10-1 \\ 9-1 \\ 7-0 \end{pmatrix}$ $=\begin{pmatrix} 9 \\ 8 \\ 7 \end{pmatrix}$
3. #### Insert

$\text{g: } \vec{x} = \vec{OA} + r \cdot \vec{AB}$

$\text{g: } \vec{x} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + r \cdot \begin{pmatrix} 9 \\ 8 \\ 7 \end{pmatrix}$