Answer:
(x, y, z) = (13/7, 19/7, 25/7)
Step-by-step explanation:
You know that the vector whose components are the coefficients of the equation of the plane is perpendicular to the plane. That is (1, 2, 3) is a vector perpendicular to the plane.
The parametric equation for a line through (1, 1, 1) with this direction vector is ...
(x, y, z) = (1, 2, 3)t +(1, 1, 1) = (t+1, 2t+1, 3t+1)
The point of intersection of this line and the plane will be the point in the plane closest to (1, 1, 1). That point has a t-value of ...
(t +1) +2(2t +1) +3(3t +1) = 18
14t +6 = 18
t = 12/14 = 6/7
The point in the plane closest to (1, 1, 1) is ...
(x, y, z) = (6/7+1, 2(6/7)+1, 3(6/7)+1)
(x, y, z) = (13/7, 19/7, 25/7)
