Show that the lines and are intersecting. Hence, find their point of intersection.

Given: – Two lines having vector notion and

To show: - Lines are intersecting

The position vectors of arbitrary points on the given lines are

1^st line

2^nd line

If the lines intersect, then they must have a common point.

Therefore for some value of λ and μ, we have

⇒ (λ + 3) = 3μ + 5, 2λ + 2 = 2μ – 2, 2λ – 4 = 6μ

⇒ λ = 3μ + 2 …… (i)

⇒ λ – μ = – 2 ……(ii)

and λ – 3μ = 2 ……(iii)

putting value of λ from eq i in eq ii, we get

⇒ λ – μ = – 2

⇒ 3μ + 2 – μ = – 2

⇒ 2μ = – 4

⇒ μ = – 2

Now putting value of μ in eq i, we get

⇒ λ = 3μ + 2

⇒ λ = 3( – 2) + 2

⇒ λ = – 6 + 2

⇒ λ = – 4

As we can see by putting value of λ and μ in eq iii, that it satisfy the equation.

Check

⇒ λ – 3μ = 2

⇒ – 4 – 3( – 2) = 2

⇒ – 4 + 6 = 2

⇒ 2 = 2

⇒ LHS = RHS ; Hence intersection point exists or line do intersect

We can find an intersecting point by putting values of μ or λ in any one general point equation

Thus,

Intersection point

⇒

⇒

⇒

Hence, Intersection point is ( – 1, – 6, – 12)