Finding the Minimal Sum Unit Vector in R^3

In summary, the problem is to find the unit vector u=[x,y,z]^T in R^3 for which the sum x+8y+2z is minimal. This can be done by considering the plane x+8y+2z = c for different values of c, with the constraint that x^2+ y^2+ z^2= 1. By finding the direction that leads to a single intersection of the plane and sphere, the problem can be solved without using the Lagrange multiplier method.
  • #1
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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! :)
 
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  • #2
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)
 
  • #3
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.
 
  • #4
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
 

What is a unit vector?

A unit vector is a vector with a magnitude of 1 and is typically used to represent a direction in space. It is often denoted by a lowercase letter with a hat on top (e.g. ^v).

How do you find the unit vector?

To find the unit vector, you need to divide the original vector by its magnitude. This will result in a vector with the same direction as the original, but with a magnitude of 1.

Why is finding the unit vector important?

Finding the unit vector is important because it allows us to represent a direction without being affected by the magnitude of the vector. This makes it easier to perform calculations and analyze data.

What is the difference between a unit vector and a normal vector?

A unit vector has a magnitude of 1, while a normal vector can have any magnitude. Additionally, a unit vector is often used to represent a direction, while a normal vector is used to represent a line or surface that is perpendicular to another vector.

Can a zero vector be a unit vector?

No, a zero vector (a vector with all components equal to 0) cannot be a unit vector because its magnitude is 0. A unit vector must have a magnitude of 1.

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