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Lagrange Multipliers II

December 6, 2007

As I head out for vacation, here is a simple question.  What is the distance from the plane

ax + by + cz = d

to the origin?  The answer is a simple click away (Mathworld).  For fun, suppose we also wanted to know the coordinates of the point on the plane closest to the origin.  This can be solved as a constrained optimization problem, serving as another example of the technique of Lagrange multipliers.

Informally, we wish to minimize x2 + y2 + z2 subject to the constraint ax + by + cz – d = 0. Relax the constraint into the objective and form the function,

L(x,y,z) = x2 + y2 + z2 – λ(ax + by + cz – d) .

The stationary points are determined from

δL/δx = 2x – aλ = 0 , or x = aλ/2 [1]

δL/δy = 2y – bλ = 0 , or y = bλ/2 [2]

δL/δz = 2z – cλ = 0 , or z = cλ/2 [3]

δL/δλ = 0 implies the original constraint, ax + by + cz = d [4]

Subsititue x, y, and z from [1-3] into [4] to obtain λ = 2d/(a2 + b2 + c2) [5].

Substitute the λ value from [5] into [1-3] to obtain the coordinates of the point closest to the origin,

e = a2 + b2 + c2

x*= ad/e, y*= bd/e, z*= cd/e . The distance from (x*, y*, z*) to the origin is

D = [((ad)2 + (bd)2 + (cd)2)/e2]1/2

= d(e/e2)1/2 or D = d/√(a2 + b2 + c2), which matches the result from analytic geometry. Of course, calculus is not necessary (or even convenient) for this derivation, but it is another good illustration of Lagrange multipliers.

Formally proving that the stationary point represents a minimum is well beyond a blog post, but intuitively you can imagine that the distance from the origin to any point on the plane can be made arbitrarily large.  The stationary point can not represent a maximum distance.

Categories: Math
1. December 7, 2007 at 12:18 am

Great explanation in this post and the last. I think that’s the clearest I’ve ever seen them presented.

2. December 7, 2007 at 9:02 pm

Glad you liked it – I’ll try to post a couple more examples in the coming days. Things are moving slowly while I’m on vacation. This is a topic, however, that I’ve found people absorb the theory better after seeing many examples of the mechanics.

3. December 8, 2007 at 7:06 am

Dude – you should have been a calc. teacher. Mine didn’t event cover this. I do like seeing some examples before digging into theory.