Point on the line?

This is to check whether a point lies on a given line.

Example

Is the point $A(-3|14|10)$ located on the line $g$?.

$\text{g: } \vec{x} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + r \cdot \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}$

1. Insert $A$ in $g$

   The position vector (vector with coordinates of the point) of $A$ is used for $\vec{x}$ in $g$.

   $\begin{pmatrix} -3 \\ 14 \\ 10 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + r \cdot \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}$

2. Set up equation system

   Now we set up the equation system and solve it. Every row is an equation.
   1. $-3=3-3r$
   2. $14=4+5r$
   3. $10=6+2r$
   
   
   1. $r=2$
   2. $r=2$
   3. $r=2$

3. Check

   If there is no contradiction and $r$ is equal in all equations, then the point is on the line.

   I, II, III: $r=2$

   => Point $A$ is located on the line.