Math ↠ Relative position of lines ↠ Parallel lines

Parallel lines

Parallel lines are divided into parallel and coincident lines.

For determining if lines in three-dimensional 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.

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Requirements

  1. The direction vectors are collinear
  2. 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.

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Requirements

  1. Direction vectors are collinear
  2. The terminal point of the position vector of one line is not on the other line
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Info

Basically, for the second requirement you can take any point of one line and check if it is on the other one (in both cases). However, since the position vector is already given, you can easily use it.

Example

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

$\text{h: } \vec{x} = \begin{pmatrix} -3 \\ 14 \\ 10 \end{pmatrix} + s \cdot \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}$

  1. 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.

  2. 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.

    1. $-3=3-3r$
    2. $14=4+5r$
    3. $10=6+2r$

    1. $r=2$
    2. $r=2$
    3. $r=2$

    => The lines are coincident.