Results 1 to 9 of 9

Math Help - Line of Intersection between two planes

  1. #1
    Junior Member
    Joined
    Apr 2011
    Posts
    56

    Line of Intersection between two planes

    How would I determine the line of intersection between plane 1: 4x + 2y + 6z -14 = 0 and plane 2: x - y + z - 5 = 0??

    I don't remember exactly how to do it, and trying to find an explanation online just confused me even more

    I was thinking somewhere along the lines of using the cross products of the normals, but I'm not sure what to do after that.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,615
    Thanks
    1582
    Awards
    1

    Re: Line of Intersection between two planes

    Quote Originally Posted by IanCarney View Post
    How would I determine the line of intersection between plane 1: 4x + 2y + 6z -14 = 0 and plane 2: x - y + z - 5 = 0??
    I was thinking somewhere along the lines of using the cross products of the normals
    That cross product is now the direction vector for the line of intersection.

    Now we need to find some point that is on both planes.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2011
    Posts
    56

    Re: Line of Intersection between two planes

    Quote Originally Posted by Plato View Post
    That cross product is now the direction vector for the line of intersection.

    Now we need to find some point that is on both planes.
    Cross product would be (8, 2, -6), right?

    For the point, are you supposed to let a variable equal zero and then find the other two variables?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,615
    Thanks
    1582
    Awards
    1

    Re: Line of Intersection between two planes

    So far, so good.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Apr 2011
    Posts
    56

    Re: Line of Intersection between two planes

    Okay so assuming I let z=0

    4x+2y=14 and x-y=5

    y=-1 and x=-4

    Would the equation of the line be x=8t-4, y=2t-1, and z=-6t?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Apr 2011
    Posts
    56

    Re: Line of Intersection between two planes

    Ugh, according to the book the answer is x=8+4t, y=t, z=-3-3t

    Not sure where I went wrong :s
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,615
    Thanks
    1582
    Awards
    1

    Re: Line of Intersection between two planes

    Quote Originally Posted by IanCarney View Post
    Ugh, according to the book the answer is x=8+4t, y=t, z=-3-3t
    Those are different ways to write the same line.
    Can you prove that?


    EDIT: there may be a sign error in your point. Check the x value.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Apr 2011
    Posts
    56

    Re: Line of Intersection between two planes

    Quote Originally Posted by Plato View Post
    Those are different ways to write the same line.
    Can you prove that?


    EDIT: there may be a sign error in your point. Check the x value.
    Which x value are you talking about?

    Edit: Oh right, it's supposed to be +4. Thanks for your help!
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,719
    Thanks
    634

    Re: Line of Intersection between two planes

    Hello, IanCarney!

    I divided the first equation by 2 . . . Don't know why "they" didn't.

    Here is a rather primitive method.


    Determine the line of intersection of planes: . \begin{array}{cccc} 2x + y + 3z &=& 7 & [1] \\ x - y + z &=& 5 & [2]\end{array}

    Add [1] and [2]: . 3x + 4z \:=\:12 \quad\Rightarrow\quad x \:=\:4 - \tfrac{4}{3}z

    Substitute into [1]: . 2(4 - \tfrac{4}{3}z) + y + 3z \:=\:7 \quad\Rightarrow\quad y \:=\:\text{-}1 - \tfrac{1}{3}x

    We have: . \begin{Bmatrix}x &=& 4 - \frac{4}{3}z \\ \\[-4mm]y &=& \text{-}1 - \frac{1}{3}z \\ \\[-4mm] z &=& z \end{Bmatrix}


    On the right, replace z with the parameter t.

    . . . . . . . . . \begin{Bmatrix}x &=& 4 - \frac{4}{3}t \\ \\[-4mm] y &=& \text{-}1 - \frac{1}{3}t \\ \\[-4mm] z &=& t \end{Bmatrix}


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    We can check our results.


    Substitute our answers into [1]:

    . . . . . . . . . . . 2x + y + 3z \;=\;7

    . . 2(4-\tfrac{4}{3}) + (\text{-}1 - \tfrac{1}{3}z) + 3t \:=\:7

    . . . . . 8 - \tfrac{8}{3}y -1 - \tfrac{1}{3}t + 3t \;=\;7

    . - - - - . . . . . . . . . . . . . 7 \;=\;7 \;\;\;check!


    Substitute our answers into [2]:

    . . . . . . . . . . . . x - y + z \;=\;5

    . . (4-\tfrac{4}{3}t) - (\text{-}1-\tfrac{1}{3}t) + t \;=\;5

    . . . . . 4 - \tfrac{4}{3}t + 1 + \tfrac{1}{3}t + t \;=\;5

    . . . . . . . . . . . . . . . . .  5 \;=\;5 \;\;\;check!

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Angle between planes and line of intersection of planes.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 6th 2011, 12:08 PM
  2. Line of intersection of planes
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 5th 2010, 03:48 PM
  3. Line of intersection of two planes
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 7th 2010, 02:00 AM
  4. Line of Intersection of Planes
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 9th 2008, 05:25 PM
  5. line of intersection of planes
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 16th 2008, 03:39 AM

Search Tags


/mathhelpforum @mathhelpforum