Answer :

Given: The equations of the two lines are


and


To Prove: The two lines intersect and to find their point of intersection.


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 + 2, 4λ1 + 3)


A point on the second line is (5λ2 + 4, 2λ2 + 1, λ2)


If they intersect they should have a common point.


1 + 1 = 5λ2 + 4 1 – 5λ2 = 3 … (1)


1 + 2 = 2λ2 + 1 1 - 2λ2 = -1 … (2)


Solving (1) and (2),


-11λ2 = 11


λ2 = -1


Therefore, λ1 = -1


Substituting for the z coordinate, we get


1 + 3 = -1 and λ2 = -1


So, the lines intersect and their point of intersection is (-1, -1, -1)


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