More fun with lagrange multipliers

[hiigaranace]

Homework Statement

Find the point closest to the origin on the line of intersection of the planes y + 2z = 12 and x + y = 6

Homework Equations

\nuf = \lambda\nug1 +\mu\nug2
f = x²+y²+z²
g1: y + 2z = 12
g2: x + y = 6

There are supposed to be gradients on all of those, whether or not LaTeX wants to show them.

The Attempt at a Solution

Let \nuf(x, y, z) = 2x\vec{i}+2y\vec{j}+2z\vec{k}, \nug1(x, y, z) = \vec{j}+2\vec{k}, and \nug2(x, y, z) = \vec{i}+\vec{j}

this gives:

2x = \mu 2y = \lambda + \mu 2z = 2\lambda

I tried pushing ahead from here, but I end up getting nowhere. Can someone please help me?
...annnnnnnnnnnd much as I hate to admit it, I'm having a lot of trouble with lagrange multipliers in the first place, and my textbook is sadly not a whole lot of help. If anyone out there can explain how to work these out in a general sense, I would very much appreciate it.

 

[fzero]

It looks good so far (you probably should put some commas into make your solution easier to read). You should apply the constraints to the solution to determine the Lagrange multipliers.

 

[hiigaranace]

It looks good so far (you probably should put some commas into make your solution easier to read). You should apply the constraints to the solution to determine the Lagrange multipliers.

Soooo...plug the lambda and mu back into the equation I'm trying to optimize?

EDIT: When I try to trn the constraints into lambda and mu, I wind up with:

2\mu + \lambda -12 = 0 3\lambda+\mu-12=0

combine them, and you get

\mu = 2\lambda

so, x = 1/2\mu = \lambda = z y = 3/2 \lambda

which ends up giving me 5\lambda = 12.

Problem is that this is wrong. It's supposed to come out as point = (2, 4, 4). I don't understand what I'm doing wrong.

 

[hotvette]

You have 3 equations and 5 unknowns. You need 2 additional equations relating the unknowns - the constraint equations.

 

[hiigaranace]

You have 3 equations and 5 unknowns. You need 2 additional equations relating the unknowns - the constraint equations.

I'm sorry, I don't follow. I thought the constraint equations were:

g1=y+2z=12
g2=x+y=6

The other equations I have are:

f = x²+y²+z²
\grad{f} = 2x+2y+2z
\grad{g1} = \vec{j} + 2\vec{k}
\grad{g2} = \vec{i} + \vec{j}

if these are what you had in mind, then I'm afraid I don't see how to go forward without running into the same problems as before.

 

[fzero]

Soooo...plug the lambda and mu back into the equation I'm trying to optimize?

EDIT: When I try to trn the constraints into lambda and mu, I wind up with:

2\mu + \lambda -12 = 0 3\lambda+\mu-12=0

There's a mistake in your 2nd equation, it should be 2\mu + \lambda =12. I find (2,4,4). Just recheck your algebra.

 

[hiigaranace]

Ah, crud, I see it now. Blasted algebra always gets me.
Thanks for the help!