Results 1 to 2 of 2

Math Help - Intersection of Planes

  1. #1
    Senior Member
    Joined
    Nov 2008
    Posts
    425

    Intersection of Planes

    State whether each of the following pairs of planes intersect. If the planes do intersect, determine the eqtn of their line of intersection.

    x-y+z-2=0
    2x+y+z-4=0


    my work:
    eqtn 1 + eqtn 2
    3x+2z-6=0
    x=-2/3t +2

    let z=t

    -2/3t + 2 -y + t -2 =0
    y=1/3 t
    z=t

    therefore parametric equation of line is:
    x=2- 2/3 t
    y=1/3 t
    z=t

    However, the answer in the back of the book has:

    x=2-2t
    y=t
    z=3t

    How do I get this answer? i think there's just one step I am missing to convert it to that that I am not getting.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member redsoxfan325's Avatar
    Joined
    Feb 2009
    From
    Swampscott, MA
    Posts
    943
    Quote Originally Posted by skeske1234 View Post
    State whether each of the following pairs of planes intersect. If the planes do intersect, determine the eqtn of their line of intersection.

    x-y+z-2=0
    2x+y+z-4=0


    my work:
    eqtn 1 + eqtn 2
    3x+2z-6=0
    x=-2/3t +2

    let z=t

    -2/3t + 2 -y + t -2 =0
    y=1/3 t
    z=t

    therefore parametric equation of line is:
    x=2- 2/3 t
    y=1/3 t
    z=t

    However, the answer in the back of the book has:

    x=2-2t
    y=t
    z=3t

    How do I get this answer? i think there's just one step I am missing to convert it to that that I am not getting.
    Replace t with 3t. (Your value for t can arbitrary, so long as it's linear.)

    ----------

    A way to get directly to the answer in the book is as follows:

    Set the equations equal to each other and simplify:

    x-y+z-2=2x+y+z-4 \implies x+2y-2=0 \implies x=2-2y

    Let y=t. Thus,

    x=2-2t and y=t. To find z we need to sub back into one of the equations; we'll use the first one:

    2-2t-t+z-2=0 \implies z=3t

    So the (parametric) equation of the line is:

    \langle 2-2t,t,3t\rangle=\langle 2,0,0\rangle+t\langle -2,1,3\rangle

    (which, in my opinion, is a better way to write it, because it gives you the direction vector of the line).
    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. Intersection of three planes
    Posted in the Calculus Forum
    Replies: 15
    Last Post: September 25th 2009, 03:28 PM
  3. Intersection of 3 planes
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 13th 2009, 01:09 PM
  4. Intersection of 3 planes
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 13th 2009, 09:06 AM
  5. intersection of planes
    Posted in the Geometry Forum
    Replies: 1
    Last Post: June 4th 2009, 07:45 PM

Search Tags


/mathhelpforum @mathhelpforum