Finding the Minimal Sum Unit Vector in R^3

[Luxe]

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! :)

 

[lanedance]

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)

 

[HallsofIvy]

The problem is simply to minimize x+ 8y+ 2z= 0 with the constraint $x^2+ y^2+ z^2= 1$.

"Lagrange multiplier method" seems in order.

 

[lanedance]

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