example of using Lagrange multipliers


One to determine the perpendicular distance of the parallel planesMathworldPlanetmath

Ax+By+Cz+D= 0andAx+By+Cz+E= 0

is to use the Lagrange multiplier method.  In this case we may to minimise the Euclidean distance of a point  (x,y,z)  of the former plane to a (fixed) point  (x0,y0,z0)  of the latter plane.

Thus we have the equation  Ax0+By0+Cz0+E= 0  which we can subtract from the first plane equation, getting

g:=A(x-x0)+B(y-y0)+C(z-x0)+D-E= 0. (1)

This is the (only) constraint equation for minimising the square (http://planetmath.org/SquareOfANumber)

f:=(x-x0)2+(y-y0)2+(z-x0)2 (2)

of the distanceMathworldPlanetmath of the points.

The polynomial functions f and g satisfy the differentiability requirements.  Accordingly, we can find the minimising point  (x,y,z)  by considering the system of equations formed by (1) and

{fx+λgx 2(x-x0)+λA= 0,fy+λgy 2(y-y0)+λB= 0,fz+λgz 2(z-z0)+λC= 0. (3)

We solve from (3) the differences

x-x0=-Aλ2,y-y0=-Bλ2,z-z0=-Cλ2

and set them into (1).  It then yields the value

λ=2(D-E)A2+B2+C2

of the Lagrange multiplier, which we substitute into the preceding three equations obtaining

x-x0=A(D-E)A2+B2+C2,y-y0=B(D-E)A2+B2+C2,z-z0=C(D-E)A2+B2+C2.

These values give the minimal distance when put into the expression of f:

d=(D-E)2(A2+B2+C2)(A2+B2+C2)2.

Hence we have gotten the distance

d=|D-E|A2+B2+C2.
Title example of using Lagrange multipliers
Canonical name ExampleOfUsingLagrangeMultipliers
Date of creation 2013-03-22 18:48:12
Last modified on 2013-03-22 18:48:12
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 51N20
Classification msc 26B10
Synonym example of Lagrange multipliers
Related topic ParallelismOfTwoPlanes
Related topic ExampleNeedingTwoLagrangeMultipliers