Show that the lines →r = 3^i + 2^j + 4^k + λ(^i + 2^j + 2^k) and →r = 5^i + 2^j + μ(3^i + 2^j + 6^k) are intersecting. Hence find their point of intersection.

Let the lines be
M: →r = 3→i + 2→j + 4→k + λ(→i + 2→j + 2→k)
and
N: →r = 5→i + 2→j + μ(3→i + 2→j + 6→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 = 2, μ = 0
Substitute the values in equation 3,
4 + 2(2) = 6(0)
8 = 0
This is a contradiction. Therefore, there is an error in the problem statement or the provided solution is incorrect. Let's reconsider the equations.
From equation (2), we have 2λ = 2μ which simplifies to λ = μ.
Substituting λ = μ into equation (1), we get 3 + λ = 5 + 3λ, which gives 2λ = -2, so λ = -1.
Since λ = μ, we have μ = -1.
Now substitute λ = -1 into equation (3):
4 + 2(-1) = 6μ
2 = 6μ
μ = 1/3
This is a contradiction since we found μ = -1 previously. The lines do not intersect.
Let's assume there was a mistake in the original question and re-examine the equations. There seems to be an inconsistency in the given solution. A correct solution requires a consistent set of equations that can be solved simultaneously to find the point of intersection (if it exists).