# How do I know if two lines are parallel in three-dimensional space?

Oct 19, 2014

Great question, because in space two lines that "never meet" might not be parallel. Those would be skew lines, like a freeway and an overpass.

To check for parallel-ness (parallelity?) we can choose two points on each line (depending on how the lines and equations are presented), then for each pair of points, subtract the coordinates to get the displacement vector. If the two displacement or direction vectors are multiples of each other, the lines were parallel.

If your lines are given in the "double equals" form

$L : \frac{x - {x}_{o}}{a} = \frac{y - {y}_{o}}{b} = \frac{z - {z}_{o}}{c}$ the direction vector is $\left(a , b , c\right) .$

Example: Say your lines are given by equations:

${L}_{1} : \frac{x - 3}{5} = \frac{y - 1}{2} = \frac{z}{- 1}$

${L}_{2} : \frac{x - 8}{- 10} = \frac{y + 6}{- 4} = \frac{z - 2}{2}$

These lines are parallel since the direction vectors are

${v}_{1} = \left(5 , 2 , - 1\right)$ and ${v}_{2} = \left(- 10 , - 4 , 2\right)$

and it's easy to see that ${v}_{2} = - 2 {v}_{1.}$

If your lines are given in parametric form, its like the above:

${L}_{1} : x = 3 + 5 t \mathmr{and} y = 1 + 2 t \mathmr{and} z = 0 - t$

${L}_{2} : x = 8 - 10 u \mathmr{and} y = - 6 - 4 u \mathmr{and} z = 2 + 2 u$

Find the (same) direction vectors as before and see if they are scalar multiples of each other.

This space-y answer was provided by \ dansmath /