distance of non-parallel lines

As an application of the vector product (http://planetmath.org/CrossProduct) we derive the expression of the $d$ between two non-parallel lines in ${ℝ}^3$.

Suppose that the position vectors of the points of the two non-parallel lines are expressed in parametric forms

$$\overrightarrow{r}=\overrightarrow{a}+s\overrightarrow{u}$$

and

$$\overrightarrow{r}=\overrightarrow{b}+t\overrightarrow{v},$$

where $s$ and $t$ are parameters.  A common vector of the lines is the cross product $\overrightarrow{u}\times \overrightarrow{v}$ of the direction vectors of the lines, and it may be normed to a unit vector

$$\overrightarrow{n}:=\frac{\overrightarrow{u}\times \overrightarrow{v}}{|\overrightarrow{u}\times \overrightarrow{v}|}$$

by dividing it by its , which is distinct from 0 because of the non-parallelity.  The vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are the position vectors of certain points $A$ and $B$ on the lines, and thus their difference $\overrightarrow{a}-\overrightarrow{b}$ is the vector from $B$ to $A$.  If we project $\overrightarrow{a}-\overrightarrow{b}$ on the unit normal $\overrightarrow{n}$, the obtained vector

$$\overrightarrow{d}:=[(\overrightarrow{a}-\overrightarrow{b})\cdot \overrightarrow{n}]\overrightarrow{n}$$

has the sought   $d=|(\overrightarrow{a}-\overrightarrow{b})\cdot \overrightarrow{n}|$,  i.e.

$$d=\frac{|(\overrightarrow{a}-\overrightarrow{b})\cdot (\overrightarrow{u}\times \overrightarrow{v})|}{|\overrightarrow{u}\times \overrightarrow{v}|}.$$

For illustrating that $d$ is the minimal distance between points of the two lines we underline, that the construction of $d$ guarantees that it connects two points on the lines and is perpendicular to both lines — thus any displacement of its end point makes it longer.

Notes.  The numerator is the absolute value of a triple scalar product.  If the lines intersect each other, then the connecting vector $\overrightarrow{a}-\overrightarrow{b}$ is at right angles to the common normal vector $\overrightarrow{n}$ of their plane and thus the dot product of these vectors vanishes, i.e. also  $d=0$.  If the lines do not intersect, they are called agonic lines or skew lines;  then  $d>0$.

Title	distance of non-parallel lines
Canonical name	DistanceOfNonparallelLines
Date of creation	2013-03-22 15:27:16
Last modified on	2013-03-22 15:27:16
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Derivation
Classification	msc 15A72
Synonym	distance of lines
Related topic	LineInSpace
Related topic	DistanceFromPointToALine
Related topic	EuclideanDistance
Related topic	AngleBetweenTwoLines
Defines	agonic lines
Defines	skew lines