Given: - Line joining A(0, – 1, – 1) and B(4, 5, 1).

Line joining C(3, 9, 4) and D( – 4, 4, 4).

To Prove: - Both lines intersects

Proof: - Equation of a line joined by two points A(x₁,y₁,z₁) and B(x₂,y₂,z₂) is given by

Now equation of line joining A(0, – 1, – 1) and B(4, 5, 1)

=

⇒ x = 4λ, y = 6λ – 1 and z = 2λ – 1

So, the coordinates of a general point on this line are

(4λ, 6λ – 1, 2λ – 1)

And equation of line joining C(3, 9, 4) and D( – 4, 4, 4)

=

⇒ x = – 7μ + 3, y = – 5μ + 9 and z = 4

So, the coordinates of a general point on this line are

( – 7μ + 3, – 5μ + 9, 4)

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

Therefore for some value of λ and μ, we have

⇒ 4λ = – 7μ + 3 , 6λ – 1 = – 5μ + 9, and 2λ – 1 = 4

⇒ ……(i)

⇒ 6λ + 5μ = 10 ……(ii)

and 2λ = 5 ……(iii)

from eq iii, we get

⇒

Now putting the value of λ in eq i, we get

⇒

⇒ – 1

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

Check

⇒ 6λ + 5μ = 10

⇒

⇒

⇒ 10 = 10

⇒ LHS = RHS ;Hence intersection point exist or line do intersects

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

Thus,

Intersection point

– 7μ + 3, – 5μ + 9, 4

– 7( – 1) + 3, – 5( – 1) + 9, 4

10, 14, 4