Thursday

October 23, 2014

October 23, 2014

Posted by **Justin** on Tuesday, December 7, 2010 at 9:46pm.

- Math -
**MathMate**, Tuesday, December 7, 2010 at 11:54pmRead up your textbook or lecture notes on Lagrange multipliers. It is not as difficult as it sounds.

However, you need to master you algebra and basic calculus, which I believe should not be a problem.

To put it in the least words possible, we attempt to optimize (maximize or minimize) a function f(x,y,z) subject to the constraint g(x,y,z).

The suggested objective function is the square of the distance from the origin, which therefore is

f(x,y,z)=x²+y²+z²

The constraint is

g(x,y,z)=x-2y+3z-14=0

We now introduce a Lagrange multiplier, λ, to form a new function Λ:

Λ(x,y,z)=f(x)-λg(x)

Λ(x,y,z)=x²+y²+z²-λ(x-2y+3z-14)

Now apply partial differentiation with respect to each of the variables, and equate result to zero:

∂Λ/∂x=2x-λ...(1)

∂Λ/∂y=2y+2λ...(2)

∂Λ/∂z=2z-3λ...(3)

Add the original constraint equation:

x-2y+3z-14=0...(4)

Now solve the system of 4 equations in x,y,z and λ and voilą!

x=1, y=-2, z=3, L=2.

So the distance is

D=√(1²+(-2)²+3²)

=√14

How can we tell if this is correct?

This is not too difficult... in this particular case.

The shortest distance from a point (origin 0,0,0) to a plane is the perpendicular distance, given by the well-known formula:

Dmin=(ax0+by0+cz0+d)/√(a²+b*sup2;+c²)

=(0-0+0-14)/√(1²+(-2)²+3²)

=-14/√(14)

=-√(14)

and the square of the minimum distance

Dmin²=14

which checks with our Lagrange multiplier answer.

**Answer this Question**

**Related Questions**

Linear Algebra - In 3-space, a plane abc is spanned by three points a, b, c. The...

Math - How do you: Find the shortest distance, to two decimal places from the ...

calculus - Use Lagrange multipliers to find the max/min values of the function f...

7th grade math Ms. Sue please - 2. Shelly sews a blanket that has an area of 170...

Geometry - Martin wants to use coordinate geometry to prove that the opposite ...

????Math Question????8th grade adv.math - Algebra! How is the distance formula ...

maths - geometry - In a space with an orthonormal coordinate system consider the...

physics - A toy plane p starts flying from point A along a straight horizontal ...

geometry - A point is moving in a plane in such a way that its distance from the...

trigonometry - The angle of elevation of the top of a tower to point A on the ...