Q. 12

Prove that the lines and intersect each other and find the point of their intersection.

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 (λ1 + 4, 4λ1 – 3, 7λ1 – 1)


A point on the second line is (2λ2 + 1, -3λ2 – 1, 8λ2 – 10)


If they intersect they should have a common point.


λ1 + 4 = 2λ2 + 1 λ1 – 2λ2 = -3 … (1)


1 – 3 = -3λ2 – 1 1 + 3λ2 = 2 … (2)


Solving (1) and (2),


11λ2 = 14



Therefore,


Substituting for the z coordinate, we get


and


So, the lines do not intersect.


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