Answer :

Given: Points A(0, -1, -1), B(4, 5, 1), C(3, 9, 4) and D(-4, 4, 4)


To Prove: Line through A and B intersects the line through C and D.


Proof: We know that, the equation of a line passing through two points (x1, y1, z1) and (x2, y2, z2) is,



So,


The equation of line passing through points A(0, -1, -1) and B(4, 5, 1) is



Where, x1 = 0, y1 = -1 and z1 = -1


And, x2 = 4, y2 = 5 and z2 = 1




Let




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


Take .


x = 4λ


Take .


y + 1 = 6λ


y = 6λ – 1


Take


z + 1 = 2λ


z = 2λ – 1


This means, any point on the line L1 is (4λ, 6λ – 1, 2λ – 1).


The equation of line passing through points C(3, 9, 4) and D(-4, 4, 4) is



Where, x1 = 3, y1 = 9 and z1 = 4


And, x2 = -4, y2 = 4 and z2 = 4



Let




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


Take .


x – 3 = -7μ


x = -7μ + 3


Take .


y – 9 = -5μ


y = -5μ + 9


Take .


z – 4 = 0


z = 4


This means, any point on the line L2 is (-7μ + 3, -5μ + 9, 4).


If lines intersect then there exist a value of λ, μ for which


(4λ, 6λ – 1, 2λ – 1) ≡ (-7μ + 3, -5μ + 9, 4)


4λ = -7μ + 3 …(i)


6λ – 1 = -5μ + 9 …(ii)


2λ – 1 = 4 …(iii)


From equation (iii), we get


2λ – 1 = 4


2λ = 4 + 1


2λ = 5



Putting the value of λ in equation (i), we get



2 × 5 = -7μ + 3


10 = -7μ + 3


7μ = 3 – 10


7μ = -7



μ = -1


Putting the values of λ and μ in equation (ii), we get



3 × 5 – 1 = 5 + 9


15 – 1 = 14


14 = 14


Since, these values of λ and μ satisfy equation (ii), this implies that the lines intersect.


Hence, the lines through A and B intersects line through C and D.


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