Results 1 to 6 of 6

Math Help - Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ...

  1. #1
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ...

    Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.

    Answer:
    x = -0.272727272727273
    y = -0.818181818181818
    z = 0.272727272727273

    Please show me how to do this problem!

    Any help would be appreciated!
    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

    Quote Originally Posted by s3a View Post
    Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.

    Answer:
    x = -0.272727272727273
    y = -0.818181818181818
    z = 0.272727272727273

    Please show me how to do this problem!

    Any help would be appreciated!
    Thanks in advance!
    You need to minimize x^2+y^2+z^2.

    Using the method of Lagrange multipliers,

    2x=\lambda

    2y=3\lambda

    2z=-\lambda

    Substituting for x,\ y,\ z in the equation of the plane,

    -\lambda=\lambda+9\lambda+6

    11\lambda=-6

    \lambda=-\frac{6}{11}

    x=-\frac{3}{11}=-0.272727272727273

    y=-\frac{9}{11}=-0.818181818181818

    z=\frac{3}{11}=0.272727272727273
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

    Quote Originally Posted by s3a View Post
    Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.

    Answer:
    x = -0.272727272727273
    y = -0.818181818181818
    z = 0.272727272727273

    Please show me how to do this problem!

    Any help would be appreciated!
    Thanks in advance!
    Search the minimum of the function...

    f(x,y)= x^{2} + y^{2} + z^{2}= x^{2}+y^{2}+ (x+3 y+3)^{2}

    ... respect to x and y...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  4. #4
    s3a
    s3a is offline
    Super Member
    Joined
    Nov 2008
    Posts
    597

    Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

    Sorry for the stupid question but how do I know that f(x,y) = x^2 + y^2 + z^2 is what I want to minimize?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

    Quote Originally Posted by s3a View Post
    Sorry for the stupid question but how do I know that f(x,y) = x^2 + y^2 + z^2 is what I want to minimize?
    Because x^2 + y^2 + z^2 is the square of the distance of the point (x,\ y,\ z) from the origin.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1

    Re: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is ..

    Quote Originally Posted by s3a View Post
    Question: Find the coordinates of the point (x,y,z) on the plane z = x + 3y + 3 which is closest to the origin.
    Answer:
    x = -0.272727272727273
    y = -0.818181818181818
    z = 0.272727272727273
    One does not need any calculus to solve this.
    The plane is x+3y-z=-3.
    Find out where the line <t,3t,-t> intersects that plane.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find the coordinates of the turning point
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 13th 2009, 11:47 PM
  2. Replies: 2
    Last Post: July 28th 2009, 01:00 AM
  3. Find coordinates of fixed point if...?
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 26th 2008, 06:15 AM
  4. Find Coordinates of Point B
    Posted in the Geometry Forum
    Replies: 3
    Last Post: October 4th 2008, 04:24 AM
  5. Find Coordinates of Point B
    Posted in the Geometry Forum
    Replies: 2
    Last Post: September 26th 2008, 05:00 AM

Search Tags


/mathhelpforum @mathhelpforum