Answer:
(4/3, 7/3, 5/3)
Explanation:
The line perpendicular to the plane through the point will be ...
... (x, y, z) = (1, 1, 1) +t(1, 4, 2)
Using this expression for (x, y, z), we can find the point on the line where it intersects the plane. That point is the one of interest.
... (1 +t) +4(1 +4t) +2(1 +2t) = 14 . . . . . substituting for (x, y, z) in the equation of the plane
... 7 +21t = 14 . . . . . collect terms
... 21t = 7 . . . . . . . . subtract 7
... t = 7/21 = 1/3 . . . divide by 21
The point of interest is ...
... (x, y, z) = (1, 1, 1) + (1/3)(1, 4, 2) = (1 1/3, 2 1/3, 1 2/3)