Show that the lines

and intersect.


Also, find their point of intersection.


We are given with lines,



Let these lines be L1 and L2, such that




Where λ, μ


We need to show that lines L1 and L2 intersect.


In order to show this, let us find out any point on line L1 and line L2.


For line L1:





We need to find the values of x, y and z. So,


Take .


x – 1 = 2λ


x = 2λ + 1


Take .


y – 2 = 3λ


y = 3λ + 2


Take .


z – 3 = 4λ


z = 4λ + 3 …(i)


, any point on line L1 is (2λ + 1, 3λ + 2, 4λ + 3).


For line L2:





We need to find the values of x, y and z. So,


Take .


x – 4 = 5μ


x = 5μ + 4


Take .


y – 1 = 2μ


y = 2μ + 1


Take


z = μ …(ii)


, any point on line L2 is (5μ + 4, 2μ + 1, μ).


Since, if line L1 and L2 intersects then there exists λ and μ such that,


(2λ + 1, 3λ + 2, 4λ + 3) ≡ (5μ + 4, 2μ + 1, μ)


2λ + 1 = 5μ + 4 …(iii)


3λ + 2 = 2μ + 1 …(iv)


4λ + 3 = μ …(v)


Substituting value of μ from equation (v) in equation (iv), we get


3λ + 2 = 2(4λ + 3) + 1


3λ + 2 = 8λ + 6 + 1


3λ + 2 = 8λ + 7


8λ – 3λ = 2 – 7


5λ = -5



λ = -1


Putting λ = -1 in equation (v), we get


4(-1) + 3 = μ


μ = -4 + 3


μ = -1


To check, put λ = -1 and μ = -1 in equation (iii),


2(-1) + 1 = 5(-1) + 4


-2 + 1 = -5 + 4


-1 = -1


, λ and μ also satisfy equation (iii).


So, z-coordinate from equation (i),


z = 4λ + 3


z = 4(-1) + 3 [, λ = -1]


z = -4 + 3


z = -1


And z-coordinate from equation (ii),


z = μ


z = -1 [, μ = -1]


So, the lines intersect.


Their point of intersection is (5μ + 4, 2μ + 1, μ) = (5(-1) + 4, 2(-1) + 1, -1)


Or (5μ + 4, 2μ + 1, μ) = (-5 + 4, -2 + 1, -1)


Or (5μ + 4, 2μ + 1, μ) = (-1, -1, -1)


Thus, the given lines intersect, and the point of intersection is (-1, -1, -1).


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