. . Shortest distance between two straight lines We have just explained how the point of intersection of two lines are found and we have also studied how to determine whether the given two lines are parallel or not. Definition .
Two lines are said to be coplanar if they lie in the same plane. Note If two lines are either parallel or intersecting, then they are coplanar. Definition . Two lines in space are called skew lines if they are not parallel and do not intersect Vector - - Applications of Vector Algebra Note If two lines are skew lines, then they are non coplanar.
If the lines are not parallel and intersect, the distance between them is zero. If they are parallel and non-intersecting, the distance is determined by the length of the line segment perpendicular to both the parallel lines. In the same way, the shortest distance between two skew lines is defined as the length of the line segment perpendicular to both the skew lines. Two lines will either be parallel or skew.
Theorem . The shortest distance between the two parallel lines r sb and r tb is given by | ( , where | b ¹ . Proof The given two parallel lines r sb and r tb are denoted by L and L respectively. Let A and B be the points on L and L whose position vectors are a and c respectively.
The two given lines are parallel to b Let AD be a perpendicular to the two given lines. If θ is the acute angle between AB and b , then sin θ = | | ( | | | | AB b AB ... ( ) But, from the right angle triangle ABD , sin θ = AB AB ... ( ) From ( ) and ( ), we have d = | ( , where | b ¹ .
Theorem . The shortest distance between the two skew lines r sb and r td is given by δ = | ( ) (