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.

2λ₁ + 1 = 5λ₂ + 4 ⇒ 2λ₁ – 5λ₂ = 3 … (1)

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

Solving (1) and (2),

-11λ₂ = 11

λ₂ = -1

Therefore, λ₁ = -1

Substituting for the z coordinate, we get

4λ₁ + 3 = -1 and λ₂ = -1

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