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 (0, 2, -3) and has direction ratios (1, 2, 3)


Therefore, vector equation of line L1 is



And



Line L2 is passing through point (2, 6, 3) and has direction ratios (2, 3, 4)


Therefore, vector equation of line L2 is



Now, to calculate distance between the lines,




Here,






Therefore,










Now,




= - 2 + 8 – 6


= 0


Therefore, the shortest distance between the given lines is





As d = 0


Hence, given lines intersect each other.


Now, general point on L1 is


x1 = λ , y1 = 2λ+2 , z1 = 3λ-3


let, P(x1, y1, z1) be point of intersection of two given lines.


Therefore, point P satisfies equation of line L2.




3λ – 6 = 4λ – 8


λ = 2


Therefore, x1 = 2 , y1 = 2(2)+2 , z1 = 3(2)-3


x1 = 2 , y1 = 6 , z1 = 3


Hence point of intersection of given lines is (2, 6, 3).


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