Given: The equations of the two lines are

and

_{To Prove: the lines do not intersect each other.}

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 – 1, λ₁)

A point on the second line is (5λ₂ - 1, λ₂ + 1, 2)

If they intersect they should have a common point.

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

3λ₁ – 1 = λ₂ + 1 ⇒ 3λ₁ - λ₂ = 2 … (2)

Solving (1) and (2),

-13λ₂ = -10

Therefore,

Substituting for the z coordinate, we get

and z = 2

So, the lines do not intersect.