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Math Help - Another whether a line is parallel to a plane?

  1. #1
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    Another whether a line is parallel to a plane?

    How can you tell whether the line x = x0 + tv in R3 is parallel to the plane x = x0 + t1v1 + t2v2 ?
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  2. #2
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    Re: Another whether a line is parallel to a plane?

    Hey Civy71.

    Hint: If a line lies on a plane then the line will be perpendicular to the plane normal.
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    Re: Another whether a line is parallel to a plane?

    Quote Originally Posted by Civy71 View Post
    How can you tell whether the line x = x_0 + tv in R3 is parallel to the plane x = x_0 + t_1v_1 + t_2v_2 ?
    Surely both line and plane have x_0 in common.

    Question is v\cdot(t_1\times t_2)=0~?
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  4. #4
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    Re: Another whether a line is parallel to a plane?

    yes, it equals 0
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    Re: Another whether a line is parallel to a plane?

    Quote Originally Posted by Civy71 View Post
    yes, it equals 0
    Well then, a line with direction vector D is parallel to a plane with normal N if and only if D\cdot N=0.
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  6. #6
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    Re: Another whether a line is parallel to a plane?

    Here's another way to view the solution, purely algebraic.

    Another whether a line is parallel to a plane?-mhflineparallelplanea.png
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    Re: Another whether a line is parallel to a plane?

    Quote Originally Posted by Civy71 View Post
    How can you tell whether the line x = x0 + tv in R3 is parallel to the plane x = x0 + t1v1 + t2v2 ?
    v.v1Xv2 = 0. Same as det |v,v1,v2 | = 0 , ie v, v1 and v2 are linearly dependent.

    EDIT: v is a vector in the direction of the line. v1 and v2 are vectors in the plane.
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