Answer :

Given: The equations of the two lines are


and


To Prove: the lines do not intersect each other.


Formula Used: Equation of a line is


Vector form:


Cartesian form:


where is a point on the line and b1 : b2 : b3 is the direction ratios of the line.


Proof:


Let




So a point on the first line is (2λ1 + 1, 3λ1 – 1, λ1)


A point on the second line is (5λ2 - 1, λ2 + 1, 2)


If they intersect they should have a common point.


1 + 1 = 5λ2 - 1 1 – 5λ2 = -2 … (1)


1 – 1 = λ2 + 1 1 - λ2 = 2 … (2)


Solving (1) and (2),


-13λ2 = -10



Therefore,


Substituting for the z coordinate, we get


and z = 2


So, the lines do not intersect.


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