Let the lines are \(M:\vec r = 3\hat i+ 2\hat j - 4\hat k + \lambda(\hat i + 2\hat j + 2\hat k)\) and \(N: \vec r = 5\hat i - 2\hat j + \,u(3\hat i + 2\hat j + 6 \hat k)\)
Coordinates of any random point on M are P(3 + λ, 2 + 2λ, −4 + 2λ) and on N are Q(5 + 3μ, −2 + 2μ, 6μ)
If the lines M and N intersect then, they must have a common point on them i.e., P and Q must coincide for some values of λ and μ
Now, 3 + λ = 5 + 3μ -----(1)
2 + 2λ = −2 + 2μ- ----(2)
−4 + 2λ = 6μ -----(3)
Solving (1) and (2),
we get λ = −4, μ = −2
Substitute the values in equation 3,
−4 + 2(−4) = 6(−2)
−4 = −12 + 8
−4 = −4
So, the given lines intersect each other
Now, point of intersection is P(−1, −6, −12).