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Math Help - Vector equation for a plane

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    Vector equation for a plane

    Hey guys, quick question. If a plane passes through the points (a,b,c) and (d,e,f), and is parallel to the vector (3,2,1). Am i right in assuming the vector equation for the plane will be (x,y,z) = (a,b,c) + [(d,e,f) - (a,b,c)]s + (3,2,1)s. Thanks
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    Bar0n janvdl's Avatar
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    Quote Originally Posted by Oiler View Post
    Hey guys, quick question. If a plane passes through the points (a,b,c) and (d,e,f), and is parallel to the vector (3,2,1). Am i right in assuming the vector equation for the plane will be (x,y,z) = (a,b,c) + [(d,e,f) - (a,b,c)]s + (3,2,1)s. Thanks
    If it passes through points (a,b,c) and (d,e,f) then t(a - d, b - e, c - f) is also a vector it is parallel to.

    (x,y,z) = (a,b,c) + t(a-d, b-e, c-f)
    (x,y,z) = (a,b,c) + t(3, 2, 1)

    I suppose you could combine them into:
    2(x,y,z) = 2(a,b,c) + t(3 + a - d, 2 + b - e, 1 + c - f)

    (x,y,z) = (a,b,c) + \frac{t(3 + a - d, 2 + b - e, 1 + c - f)}{2}


    EDIT:

    But I'm not sure why you would use such an elaborate method.
    (x,y,z) = (a,b,c) + t_1 (3,2,1)
    (x,y,z) = (d,e,f) + t_2 (3,2,1)

    You could use either one of those equations as well. If this plane passes through both points, and is parallel to a certain vector then you only need one of those points and the vector it's parallel to.
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  3. #3
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    Quote Originally Posted by Oiler View Post
    Hey guys, quick question. If a plane passes through the points (a,b,c) and (d,e,f), and is parallel to the vector (3,2,1). Am i right in assuming the vector equation for the plane will be (x,y,z) = (a,b,c) + [(d,e,f) - (a,b,c)]s + (3,2,1)t. Thanks
    Nearly! You have to use a 2nd variable. (see my correction in red)

    Quote Originally Posted by janvdl View Post
    If it passes through points (a,b,c) and (d,e,f) then t(a - d, b - e, c - f) is also a vector it is parallel to.
    That's correct!

    (x,y,z) = (a,b,c) + t(a-d, b-e, c-f)
    (x,y,z) = (a,b,c) + t(3, 2, 1)

    I suppose you could combine them into:
    2(x,y,z) = 2(a,b,c) + t(3 + a - d, 2 + b - e, 1 + c - f)

    (x,y,z) = (a,b,c) + \frac{t(3 + a - d, 2 + b - e, 1 + c - f)}{2}


    EDIT:

    But I'm not sure why you would use such an elaborate method.
    (x,y,z) = (a,b,c) + t_1 (3,2,1)
    (x,y,z) = (d,e,f) + t_2 (3,2,1)

    You could use either one of those equations as well. If this plane passes through both points, and is parallel to a certain vector then you only need one of those points and the vector it's parallel to.
    Maybe I don't understand your reply correctly but in general the equations you gave represent straight lines which are placed in the plane you are looking for.
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    Bar0n janvdl's Avatar
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    Yeah, sorry I misread the question and went looking for lines instead of planes.
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