distance of non-parallel lines
As an application of the vector product (http://planetmath.org/CrossProduct) we derive the expression of the between two non-parallel lines in .
Suppose that the position vectors of the points of the two non-parallel lines are expressed in parametric forms
by dividing it by its , which is distinct from 0 because of the non-parallelity. The vectors and are the position vectors of certain points and on the lines, and thus their difference is the vector from to . If we project on the unit normal , the obtained vector
has the sought , i.e.
For illustrating that is the minimal distance between points of the two lines we underline, that the construction of 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 is at right angles to the common normal vector of their plane and thus the dot product of these vectors vanishes, i.e. also . If the lines do not intersect, they are called agonic lines or skew lines; then .
|Title||distance of non-parallel lines|
|Date of creation||2013-03-22 15:27:16|
|Last modified on||2013-03-22 15:27:16|
|Last modified by||pahio (2872)|
|Synonym||distance of lines|