(a) It is easiest to express each point on the plane in the form
(10 − 2y − 3z, y, z) with ( y, z) a point in the yz-plane.
The function we want to minimize is the squared distance
f ( y, z) = (10 − 2y − 3z)2 + y2 + z2
Critical points are solutions of
2y − 4(10 − 2y − 3z) = 0
2z − 6(10 − 2y − 3z) = 0 .
The only solution is y = 10/7 , z = 15/7 . The corresponding x-coordinate is
x = 10 − 2 10/7 − 3 15/7 = 5/7 .
Hence the point on the plane, closest to the origin, is x81 5/7 , 10/7 , 15/7
(b) In this case the function we want to minimize is f
( y, z) = (10 − 2y − 3z − 1)2 + ( y − 1)2 + (z − 1)2
Critical points are solutions of
2y − 2 − 4(9 − 2y − 3z) = 0
2z − 2 − 6(9 − 2y − 3z) = 0
The only solution is y = 11/7 , z = 13/7 . The corresponding x-coordinate is
x = 10 − 2 11/7 − 3 13/7 = 9/7
Hence the point on the plane, closest to (1, 1, 1), is x81 9/7 , 11/7 , 13/7
