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

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)

4λ₁ – 3 = -3λ₂ – 1 ⇒ 4λ₁ + 3λ₂ = 2 … (2)

Solving (1) and (2),

11λ₂ = 14

Therefore,

Substituting for the z coordinate, we get

and

So, the lines do not intersect.