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Math Help - point lie in plane

  1. #1
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    point lie in plane

    does the point (4,5,6) lie in the plane (x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)<br />
    i started by writing the parametric equations

    x=4+3p-6q
    y=1-2p+6q
    z=6+p-q


    subbing in  (4,5,6) into the corresponding equations

    4=4+3p-6q
    5=1-2p+6q
    6=6+p-q

    simplifying gives

    0=3p-6q
    4=-2p+6q
    0=p-q thus p=q

    what do i do now? i know i need to figure out p and q i just cant figure it out
    Last edited by william; March 10th 2010 at 04:10 PM.
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  2. #2
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    Quote Originally Posted by william View Post
    does the point (4,5,6) lie in the plane (x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)<br />
    i started by writing the parametric equations

    x=4+3p-6q
    y=1-2p+6q
    z=6+p-q


    subbing in  (4,5,6) into the corresponding equations

    4=4+3p-6q
    5=1-2p+6q
    6=6+p-q

    simplifying gives

    0=3p-6q
    4=-2p+6q
    0=p-q thus p=q

    what do i do now?
    You need to check that p can equal q.

    What happens if you plug it into the other two equations?
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  3. #3
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    Quote Originally Posted by william View Post
    does the point (4,5,6) lie in the plane (x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)<br />
    i started by writing the parametric equations

    x=4+3p-6q
    y=1-2p+6q
    z=6+p-q


    subbing in  (4,5,6) into the corresponding equations

    4=4+3p-6q
    5=1-2p+6q
    6=6+p-q

    simplifying gives

    0=3p-6q
    4=-2p+6q
    0=p-q thus p=q

    what do i do now? i know i need to figure out p and q i just cant figure it out
    i really don't get how to check

    for example if i plug into the first equation

    0=3p-6p
    0=3p
    0=p
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  4. #4
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    Quote Originally Posted by william View Post
    i really don't get how to check

    for example if i plug into the first equation

    0=3p-6p
    0=3p
    0=p
    Okay so now you know that p=q=0 . This says the the point (4,5,6)=... is this true?
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  5. #5
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    Quote Originally Posted by TheEmptySet View Post
    Okay so now you know that p=q=0 . This says the the point (4,5,6)=... is this true?
    its just the answers in my book show p=4 and q=2 im confused how theyre getting that because i'm simply not getting anything close to that
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  6. #6
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    Quote Originally Posted by william View Post
    its just the answers in my book show p=4 and q=2 im confused how theyre getting that because i'm simply not getting anything close to that
    Plug p and q into what you started with


    (x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)


    (x,y,z)=(4,1,6)+4(3,-2,1)+2(-6,6,-1)


    (x,y,z)=(4,1,6)+(12,-8,4)+(-12,12,-2)=(4,5,8)

    So solution does not check. So something is wrong either in the book or in the problem wne it was written down.

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  7. #7
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    Quote Originally Posted by TheEmptySet View Post
    Plug p and q into what you started with


    (x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)


    (x,y,z)=(4,1,6)+4(3,-2,1)+2(-6,6,-1)


    (x,y,z)=(4,1,6)+(12,-8,4)+(-12,12,-2)=(4,5,8)

    So solution does not check. So something is wrong either in the book or in the problem wne it was written down.

    the book says that p=4 and q=2, as answers. these i am assuming are supposed to be obtained by substitution into the parametric equations, as we were doing earlier, but i do not know how to obtain those numbers. also, since they do not match up, does that not simply mean that the point does not lie in the plane?
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  8. #8
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    if you were wondering, i did solve the equations to find p and q, i finally figured out. i simply subtracted two of the equations using the elimination method.
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