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Math Help - Does the line contain in the plane?

  1. #1
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    Question Does the line contain in the plane?

    Give x=3-t, y=2+t, z=1-3t
    2x+2y-5=0

    a) determine whether the line and plane are parallel, perpendicular , or either.
    My answer to part a) is : the line is parallel to the given plane.

    b) Does the line contain in the plane?
    How should I answer this part?

    Thank you very much.
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  2. #2
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    Does 2\left( {3 - t} \right) + 2\left( {2 + t} \right) - 5 = 0 for every t?
    If so the line is in the plane.
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  3. #3
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    Hello, Jenny!

    Given: . \begin{array}{ccc}x\:=\:3-t \\ y\:=\:2+t \\ z\:=\:1-3t\end{array},\;\;2x + 2y - 5 \:= \:0

    (a) Determine whether the line and plane are parallel, perpendicular, or neither.
    My answer to part (a) is: the line is parallel to the given plane . . . yes

    (b) Is the line contained in the plane?

    (a) The line has direction vector: \vec{v} \:=\:\langle -1,1,-3\rangle
    The plane has normal vector: \vec{n} \:=\:\langle 2,2,0\rangle

    If the line and plane are perpendicular, then \vec{v} = k\vec{n} for some nonzero number k.
    . . This is not true . . . they are not perpendicular.

    If the line and plane are parallel, then: \vec{v}\cdot\vec{n} = 0
    . . Since \langle -1,1,-3\rangle\cdot\langle2,2,0\rangle \:=\;0 . . . they are parallel.


    (b) The equation of the plane is: . . . . . . 2x \quad+ \quad2y - 5 \:=\:0
    Substitute the parametric equations: . 2\overbrace{(3 - t)}^\downarrow + 2\overbrace{(2 + t)}^\downarrow - 5 \:=\:0

    If this statement is always true, the line lies in the plane.

    We get: . 6 - 2t + 4 + 2t - 5 \:=\:0\quad\Rightarrow\quad 5 \,=\,0\:??
    . . Hence, the line does not lie in the plane.

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