Answer :

Given : Cartesian equations of lines




To Find : i) vector equations of given lines


ii) 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 parallel lines :


The shortest distance between the parallel lines and


is given by,



Answer :


Given Cartesian equations of lines



Line L1 is passing through point (1, 2, -4) and has direction ratios (2, 3, 6)


Therefore, vector equation of line L1 is



And



Line L2 is passing through point (3, 3, -5) and has direction ratios (4, 6, 12)


Therefore, vector equation of line L2 is




Now, to calculate distance between the lines,




Here,






As , given lines are parallel to each other.


Therefore,






= 7










Therefore, the shortest distance between the given lines is





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