(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.

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.

babyslapmafro · Answer

Do I minimize x^2+y^2

amistre64 · Answer

thats not a line, its a plane

amistre64 · Answer

no, minimize a sphere at the origin

babyslapmafro · Answer

Oh right, typo

amistre64 · Answer

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

babyslapmafro · Answer

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

amistre64 · Answer

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

babyslapmafro · Answer

ok thanks

amistre64 · Answer

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

amistre64 · Answer

x+2y+z=1

t+2(2t)+t=1

6t=1, t = 1/6

the point should be (1/6, 1/3, 1/6)