Finding the Minimal Sum Unit Vector in R^3

Click For Summary

Homework Help Overview

The problem involves finding a unit vector in R^3 that minimizes the expression x + 8y + 2z, where the vector is subject to the constraint of being a unit vector.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods, including the least squares method and the Lagrange multiplier method, to approach the problem. Some question whether they are overthinking the solution, while others suggest considering geometric interpretations involving planes and spheres.

Discussion Status

There is an ongoing exploration of different methods to tackle the problem, with some participants suggesting that a geometric approach may simplify the process. No consensus has been reached, but multiple perspectives are being considered.

Contextual Notes

The problem is constrained by the requirement that the vector must be a unit vector, which adds complexity to the minimization of the expression.

Luxe
Messages
10
Reaction score
0

Homework Statement



Linear Algebra:

For all the unit vectors u=[x,y,z]^T in R^3. Find the one for which the sum x+8y+2z is minimal. (u is a 3 x 1 vector)



Homework Equations





The Attempt at a Solution



I tried working this with the least squares method...it wasn't right. I am probably overthinking this.

Any help is appreciated! :)
 
Physics news on Phys.org
all the unit vectors repressent a sphere of radius one

consider the plane x+8y+2z = c for some arbitrary c, each c representing a a different plane

you basically want to find the plane with the smallest c that still intersects the sphere (hint: which will be at only one point on the sphere... think directions)
 
The problem is simply to minimize x+ 8y+ 2z= 0 with the constraint [itex]x^2+ y^2+ z^2= 1[/itex].

"Lagrange multiplier method" seems in order.
 
lagrange is good & will work, but i think if you just consider which direction leads to a single intersection of the plane and sphere you can skip a couple of steps, though all sama sama
 

Similar threads

  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 8 ·
Replies
8
Views
2K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 2 ·
Replies
2
Views
3K
  • · Replies 6 ·
Replies
6
Views
3K
  • · Replies 9 ·
Replies
9
Views
2K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 2 ·
Replies
2
Views
1K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K