Parallel lines
Parallel lines are divided into parallel and coincident lines.
For determining if lines in threedimensional space are parallel or coincident the processing scheme can be used.
Coincident lines
Coincident lines are one and the same line with only a different equation.
Requirements
 The direction vectors are collinear
 The position vector of one line has its terminal point on the other line (Finding out if a point is on a line)
Parallel lines
Lines with parallel direction vectors which are not coincident are parallel.
Requirements
 Direction vectors are collinear
 The terminal point of the position vector of one line is not on the other line
Info
Example
$\text{g: } \vec{x} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + r \cdot \begin{pmatrix} 3 \\ 5 \\ 2 \end{pmatrix}$
$\text{h: } \vec{x} = \begin{pmatrix} 3 \\ 14 \\ 10 \end{pmatrix} + s \cdot \begin{pmatrix} 3 \\ 5 \\ 2 \end{pmatrix}$

Are the direction vectors collinear?
First, examine if the direction vectors are multiples of each other (=collinear).$\vec{a}=t\cdot\vec{b}$
$\begin{pmatrix} 3 \\ 5 \\ 2 \end{pmatrix}=t\cdot\begin{pmatrix} 3 \\ 5 \\ 2 \end{pmatrix}$
Calculate $t$ for every row
$3=t\cdot(3)$
$5=t\cdot5$
$2=t\cdot2$If $t$ equals the same in each row (here: $t=1$) then the vectors are collinear.

Insert the vector of $h$ into $g$
Now the position vector of one line (here: $h$) is inserted into the other.$\begin{pmatrix} 3 \\ 14 \\ 10 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + r \cdot \begin{pmatrix} 3 \\ 5 \\ 2 \end{pmatrix}$
Now we set up the equation system and solve it. If $r$ equals the same in every equation, then the terminal point of the position vector is on both lines and thus they are coincident.
 $3=33r$
 $14=4+5r$
 $10=6+2r$
 $r=2$
 $r=2$
 $r=2$
=> The lines are coincident.