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Math Help - Proving this line lies on this plane with a specfic equaction

  1. #1
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    Unhappy Proving this line lies on this plane with a specfic equaction

    Q.Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically.

    What I tried doing:
    I started by getting the parametric equation of (x, y, z) = (5, -4, 6) + u(1,4,-1)
    x=5+u
    y=-4+4u
    z=6-u
    I then subbed in u=0 to get a set of points
    x=5
    y=-4
    z=6

    I then got the parametric equation for (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)
    x=3+s+2t
    y=0+s-t
    z=2-s+t

    I decided to use the points (5,-4,6) and sub them into
    x=3+s+2t
    y=0+s-t
    z=2-s+t

    5=3+s+2t
    5-3=s+2t
    2=s+2t

    -4=0+s-t
    -4+0=s-t
    -4=s-t
    6=2-s+t
    6-2=-s+t
    4=-s+t

    I don't know if I'm doing this correctly. I really don't know what to so I tried that.
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  2. #2
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    All you have to do is take any two points on the line and show those points are on the plane.
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  3. #3
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    Quote Originally Posted by Plato View Post
    All you have to do is take any two points on the line and show those points are on the plane.
    How would I do that? Would I just pick any two random points?
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  4. #4
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    Quote Originally Posted by JohnBlaze View Post
    Would I just pick any two random points?
    Yes! I said any two points.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Yes! I said any two points.
    Would (1,2,3) and (4,5,6) work?
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  6. #6
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    Quote Originally Posted by JohnBlaze View Post
    Would (1,2,3) and (4,5,6) work?
    Do they belong on the line?
    The points must of course be on the line..
    Theorem: If two points of a line are on a plane then the line is a subset of the plane.
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  7. #7
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    Quote Originally Posted by JohnBlaze View Post
    How would I do that? Would I just pick any two random points?
    Choose two value for u and calculate the coordinates of the points from your parametric equations. If both points satisfy the equation of the plane, then the entire line is in the plane.
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  8. #8
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    Quote Originally Posted by HallsofIvy View Post
    Choose two value for u and calculate the coordinates of the points from your parametric equations. If both points satisfy the equation of the plane, then the entire line is in the plane.
    I chose two values for u(0 and 1)
    the sets of coordinates I got are (5,-4,6) and (6,4, 5) Is that right? How would I verify that the points satisfy the equation of the plane?
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  9. #9
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    Quote Originally Posted by Plato View Post
    Do they belong on the line?
    The points must of course be on the line..
    Theorem: If two points of a line are on a plane then the line is a subset of the plane.
    So I should pick values for u and then solve the equation?
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