Answer :

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 (12, 1, 5) and has direction ratios (-9, 4, 2)


Therefore, vector equation of line L1 is



And



Line L2 is passing through point (23, 10, 23) and has direction ratios (-6, -4, 3)


Therefore, vector equation of line L2 is



Now, to calculate distance between the lines,




Here,






Therefore,








= 65




Now,




= 220 + 135 + 1080


= 1435


Therefore, the shortest distance between the given lines is






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