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

  1. #1
    s3a
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    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!
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    MHF Contributor alexmahone's Avatar
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    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 $\displaystyle x^2+y^2+z^2$.

    Using the method of Lagrange multipliers,

    $\displaystyle 2x=\lambda$

    $\displaystyle 2y=3\lambda$

    $\displaystyle 2z=-\lambda$

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

    $\displaystyle -\lambda=\lambda+9\lambda+6$

    $\displaystyle 11\lambda=-6$

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

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

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

    $\displaystyle z=\frac{3}{11}=0.272727272727273$
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  3. #3
    MHF Contributor chisigma's Avatar
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    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...

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

    ... respect to x and y...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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  4. #4
    s3a
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    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?
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    MHF Contributor alexmahone's Avatar
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    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 $\displaystyle x^2 + y^2 + z^2$ is the square of the distance of the point $\displaystyle (x,\ y,\ z)$ from the origin.
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  6. #6
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    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 $\displaystyle x+3y-z=-3$.
    Find out where the line $\displaystyle <t,3t,-t>$ intersects that plane.
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