So given two SKEW lines (in Euclidian space) finding the shortest distance between them is fine, but how would you find the points of intersection of the two lines with the line of shortest distance (i.e. the line that is mutually perpendicular to both skew lines)?
E.g. given L1: r= (2+m)i + (3+2m)j + (4+2m)k
and L2: r= (3+f)i + (3+ f)j + (4+3f)k
(where f and m are parameters)
I believe they are skew, now how to find the line that passes through L1 and L2 and is mutually perpendicular to both? I can find the direction vector (by crossing the two other vectors) but I need a known point...
Help very appreciated!!! thanks!!!
Suppose that are two skew lines.
The distance between the two lines is .
That is rather simple formula to calculate.
However to find the points on each line that are endpoints of the shortest line segment between the lines is complicated.
The vector is perpendicular to both vectors .
Now write the equation of the plane that contains the line with normal .
That plane is .
The point of intersection of an that plane is one of the endpoints we need.
That point is .
To find the other point we use the vector we get .