Show that lines and intersect. Also find their point of intersection.

We have , and

Given lines can be written in cartesian form as

(i)

And (ii)

Let P(α ,β ,γ ) is an intersecting point of line (i) and (ii)

⇒ (α ,β ,γ ) satisfies line (i)

⇒

⇒ α = 3λ +1 , β = -λ +1 , γ = -1

Also (α ,β ,γ ) lies on line (ii)

⇒

⇒

⇒

Now,

⇒3λ -3 = 0

⇒3λ = 3

⇒ λ = 1

Also, -λ +1 = 0

⇒λ = 1

Since, the value of λ is same.

∴ both lines intersect each other at P(α ,β ,γ )

⇒ point of intersection = P(α ,β ,γ )

= (α = 3λ +1 , β = -λ +1 , γ = -1)

= (α = 3× 1 +1 , β = -1 +1 , γ = -1)

Thus , point of intersection = (4,0,-1)