Find the shortest distance between the lines given below:

and


Given : Cartesian equations of lines




To Find : distance d


Formulae :


1. Equation of line :


Equation of line passing through point A (a1, a2, a3) and having direction ratios (b1, b2, b3) is



Where,


And


2. Cross Product :


If are two vectors




then,



3. Dot Product :


If are two vectors




then,



4. Shortest distance between two lines :


The shortest distance between the skew lines and


is given by,



Answer :


Given Cartesian equations of lines



Line L1 is passing through point (1, -2, 3) and has direction ratios (-1, 1, -2)


Therefore, vector equation of line L1 is



And



Line L2 is passing through point (1, -1, -1) and has direction ratios (2, 2, -2)


Therefore, vector equation of line L2 is



Now, to calculate distance between the lines,




Here,






Therefore,










Now,




= 0 - 6 + 16


= 10


Therefore, the shortest distance between the given lines is






1