Q. Find the shortest distance between two non-intersecting lines passing through the points whose position vectors are a and b are parallel to vectors c and d respectively.
My confusion is : two non-intersecting lines must be always parallel.They must have a constant distance between them.What is the point of asking shortest distance between two lines if they are parallel?
Wikipedia says:
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel.
Does the last sentence means that if two lines are not coplanar and not intersecting,then they are not parallel?Then,do they have different distance at different points?
Yes, exactly. It's called Skew lines, which means that two lines, in three-dimensional space, does not intersect and aren't parallel.