Question

(Calc3, Lagrange multipliers) Please help me solve the following problem. Find the point on the line x+2y+z=1 that is closest to the origin.

Answer 1

Do I minimize x^2+y^2

Answer 2

thats not a line, its a plane

Answer 3

no, minimize a sphere at the origin

Answer 4

Oh right, typo

Answer 5

f(x,y,z) = x^2 + y^2 + z^2

Answer 6

Ok and if it were a line then I would minimize x^2+y^2

Answer 7

yes, distance from the origin to a line in the xy plane would represent a circle of x^2 + y^2

Answer 8

ok thanks

Answer 9

so double chk your results, define a line from the origin using the normal vector of the plane:

x = t
y = 2t
z = t

plug those into the plane equation to solve for t

Answer 10

x+2y+z=1

t+2(2t)+t=1

6t=1, t = 1/6

the point should be (1/6, 1/3, 1/6)