Given: - Two lines equation: and

To find: - Intersection point

We have,

⇒ x = 3λ – 1, y = 5λ – 3 and z = 7λ – 5

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

(3λ – 1, 5λ – 3, 7λ – 5)

The equation of the 2^nd line is

⇒ x = μ + 2, y = 3μ + 4 and z = 5μ + 6

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

(μ + 2, 3μ + 4, 5μ + 6)

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

Therefore for some value of λ and μ, we have

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

⇒ ……(i)

⇒ 5λ – 3μ = 7 ……(ii)

and 7λ – 5μ = 11 ……(iii)

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

Now putting value of μ in eq i, we get

⇒

⇒

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

Check

⇒ 7λ – 5μ = 11

⇒

⇒

⇒ 11 = 11

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

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

Thus,

Intersection point

3λ – 1, 5λ – 3, 7λ – 5