Results 1 to 2 of 2

Thread: intersection of planes

  1. #1
    Junior Member
    Joined
    Dec 2006
    Posts
    60

    intersection of planes

    1. The line of intersection of the planes 2x+y-3z=3 and x-2y+z=-1 is L. If L meets the xy plane at Point A and the z-axis at Point B, determine the length of line segment AB.

    2. Determine the Cartesian equation of the plane that is parallel to the line with equation x = -2y = 3z and that contains the line of intersection of the planes with equation x-y+z=1 and 2y-z=0

    Thanks, any help is greatly appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    12,028
    Thanks
    848
    Hello, checkmarks!

    1. The line of intersection of the planes: $\displaystyle 2x+y-3z\:=\:3\:\text{ and }\:x-2y+z\:=\:-1\:\text{ is }L.$
    If $\displaystyle L$ meets the $\displaystyle xy$-plane at point $\displaystyle A$ and the $\displaystyle z$-axis at point $\displaystyle B,$
    determine the length of line segment $\displaystyle AB.$
    First, find the equation of the plane.

    We have: .$\displaystyle \begin{array}{cccc}2x + y - 3x &=& 3 & [1] \\ x - 2y + z &=& \text{-}1 & [2] \end{array}$


    $\displaystyle \begin{array}{ccccc}\text{Multiply [1] by 2:} & 4x + 2y - 6x &=& 6 \\
    \text{Add [2]:} & x - 2y + z &=& \text{-}1 \end{array}$

    And we have: .$\displaystyle 5x - 5z \:=\:5 \quad\Rightarrow\quad x \:=\:z+1$

    Substitute into [2]: .$\displaystyle (z+1) - 2y + z \:=\:\text{-}1 \quad\Rightarrow\quad y \:=\:z+1$

    We have these equations: .$\displaystyle \begin{Bmatrix}x &=& z+1 \\ y &=& z+1 \\ z &=&z \end{Bmatrix}$

    On the right, replace $\displaystyle z$ with a parameter $\displaystyle t\!:\quad \begin{Bmatrix}x &=& t+1 \\ y &=& t+1 \\ z &=& t\end{Bmatrix}$

    These are the parametric equations of the line of intersection.


    Line $\displaystyle L$ intersects the $\displaystyle xy$-plane at $\displaystyle A$ . . . Then: .$\displaystyle z = 0.$

    Then $\displaystyle t = 0 \quad\Rightarrow\quad x = 1,\;y = 1$

    . . We have point $\displaystyle A(1,1,0)$


    Line $\displaystyle L$ intersected the $\displaystyle z$-axis at $\displaystyle B$ . . . Then: .$\displaystyle x = 0,\:y = 0$

    Then: .$\displaystyle t = \text{-1} \quad\Rightarrow\quad z \:=\:\text{-1}$

    . . We have point $\displaystyle B(0,0,\text{-}1)$


    Therefore, length of $\displaystyle AB \;=\;\sqrt{(0-1)^2 + (0-1)^2 + (-1-0)^2} \;=\;\sqrt{3}$

    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: Sep 6th 2011, 12:08 PM
  2. Intersection of three planes
    Posted in the Calculus Forum
    Replies: 15
    Last Post: Sep 25th 2009, 03:28 PM
  3. Intersection of 3 planes
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Sep 13th 2009, 01:09 PM
  4. Intersection of 3 planes
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Sep 13th 2009, 09:06 AM
  5. Intersection of two planes
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: Aug 30th 2009, 12:01 PM

Search Tags


/mathhelpforum @mathhelpforum