Results 1 to 4 of 4

Math Help - Finding if a line intersects the plane or not

  1. #1
    Junior Member
    Joined
    Jul 2012
    From
    NOVA
    Posts
    25

    Finding if a line intersects the plane or not

    If the line x = 1-4t; y = 3; z = 2t +2 intersects the plane x + 2y +2z = 5?
    I already tried plugging in the parametrization of the line into the equation of the plane, but I get an indeterminate answer!

    Any help is greatly appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,959
    Thanks
    1783
    Awards
    1

    Re: Finding if a line intersects the plane or not

    Quote Originally Posted by Giestforlife View Post
    If the line x = 1-4t; y = 3; z = 2t +2 intersects the plane x + 2y +2z = 5?
    I already tried plugging in the parametrization of the line into the equation of the plane, but I get an indeterminate answer!
    Given a line \ell: P+tD and plane \Pi: N\cdot<x-a,y-b,z-c>=0 then \ell\|\Pi if and only if  D \cdot N=0..
    That is the direction of the line is perpendicular to the normal of the plane.

    If a line is not parallel to a plane then they intersect.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2012
    From
    NOVA
    Posts
    25

    Re: Finding if a line intersects the plane or not

    Quote Originally Posted by Plato View Post
    Given a line \ell: P+tD and plane \Pi: N\cdot<x-a,y-b,z-c>=0 then \ell\|\Pi if and only if  D \cdot N=0..
    That is the direction of the line is perpendicular to the normal of the plane.

    If a line is not parallel to a plane then they intersect.
    Damn, I guess I just got tunnel visioned. But thanks so much!
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,907
    Thanks
    766

    Re: Finding if a line intersects the plane or not

    Hello, Giestforlife

    \text{Does the line }\begin{Bmatrix}x &=& 1-4t \\ y&=& 3 \\  z &=& 2t +2\end{Bmatrix}\,\text{ intersect the plane: }\,x + 2y +2z \:=\:5\,?

    I tried plugging in the parametrization of the line into the equation of the plane,
    . . but I get an indeterminate answer!
    Not sure what you mean by "indeterminate".

    We have: . (1-4t) + 2(3) + 2(2t+2) \:=\:5 \quad\Rightarrow\quad 1 - 4t + 6 + 4 + 4t \:=\:5

    and we get: . 11 \,=\,5\,?? . . . a false statement.

    This means that the line does not intersect the plane.
    . . They are parallel.


    We can check this fact.

    The direction vector of the line is:. \vec v \,=\,\langle\text{-}4,0,2\rangle
    The normal vector of the plane is:. \vec n \,=\,\langle 1,2,2\rangle

    And:. \vec v \cdot \vec n \:=\:\langle\text{-}4,0,2\rangle\cdot\langle1,2,2\rangle \:=\:\text{-}4 + 0 + 4 \:=\:0

    The line and the normal vector are perpendicular.
    Therefore, the line and plane are parallel.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 6th 2010, 06:48 AM
  2. Replies: 2
    Last Post: February 14th 2010, 12:18 PM
  3. Replies: 1
    Last Post: October 2nd 2009, 08:44 PM
  4. point where Plane P intersects
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 10th 2009, 08:51 AM
  5. Replies: 1
    Last Post: December 6th 2008, 12:07 PM

Search Tags


/mathhelpforum @mathhelpforum