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


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


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


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


Hence we have gotten the distance

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