1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Vector Calculus - Closest point to the origin

  1. May 14, 2010 #1
    1. The problem statement, all variables and given/known data

    A curve C in space is defined implicitly on the cylinder x^2+y^2=1 by the additional equation: x^2-xy+y^2-z^2=1. Find the point or points on C closest to the origin.


    2. Relevant equations

    d = ((x-x0)+(y-y0)+(z-z0))^(1/2) - This is the distance formula.
    Please note that I did NOT learn Lagrange Multipliers, yet - it is the next section in my math book.


    3. The attempt at a solution

    First, I used the distance formula: ((x-x0)^2+(y-y0)^2+(z-z0)^2)^(1/2).
    where (x0,y0,z0) = (0,0,0) - the origin.

    I solved for z from the additional equation: x^2-xy+y^2-z^2=1.

    So the equation now looks like this: ((x^2+y^2+(x^2-xy+y^2-1))^(1/2)

    I then square the entire equation (so that I can derive easier) because the extrema points remained the same if the equation were not squared.

    I solved the partials for x and y. And for some reason, I got fx=4x-y and fy=4y-x. Thus the point of intersection, which I believe to be incorrect, is (0,0).

    I plugged (0,0) back into the additional equation and I have square root by a negative number!

    And now... I am stuck.

    Can anyone please show me how to approach this problem using the derivation of the distance formula and NOT using Lagrange Multipliers?
     
  2. jcsd
  3. May 14, 2010 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    Hi number0! Welcome to PF! :smile:

    (have a square-root: √ and try using the X2 and X2 tags just above the Reply box :wink:)

    Hint: you know x2 + y2 = 1, so you really only need to minimise z2, and you can use x2 + y2 = 1 in the definition of C. :wink:
     
  4. May 14, 2010 #3
    Thanks tiny-tim!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Vector Calculus - Closest point to the origin
  1. Vector calculus (Replies: 12)

  2. Vector calculus -.- (Replies: 5)

  3. Vector calculus (Replies: 6)

  4. Vector Calculus (Replies: 1)

Loading...