Results 1 to 2 of 2

Math Help - proof for rectangular parallelepiped

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    4

    proof for rectangular parallelepiped

    Prove that if two diagonals of a rectangular parallelepiped are perpendicular, then its dimensions are equivalent to the sides of a right triangle and vice versa
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,551
    Thanks
    542
    Hello, mathwizard325!

    I've proved it one way so far . . . a vector proof.


    Prove that if two diagonals of a rectangular parallelepiped are perpendicular,
    then its dimensions are equivalent to the sides of a right triangle and vice versa.

    The "box" has dimensions: . a in the x direction,
    . . b in the y direction, c in the z direction.

    Orient it in the first octant with vertex P at the origin.

    The graph looks like this:


    Code:
            T   |                W
         (0,0,c)* - - - - - * (0,b,c)
               /|          /|
              / |         / |
       U     /  |        /  |
    (a,0,c) * - - - - - * V |
            |   |       |   |    S
            | P * - - - | - * (0,b,0)
            |  /        |  /
            | /         | /
       Q    |/          |/
    (a,0,0) * - - - - - * R
           /

    One diagonal is: . \overrightarrow{QW} \:=\:\langle\, -a,\,b,\,c\,\rangle

    . . . . Another is: . \overrightarrow{SU} \:=\:\langle\, a,\,-b,\,c\,\rangle


    Since \overrightarrow{QW} \perp \overrightarrow{SU}\!:\;\;\overrightarrow{QW}\cdot  \overrightarrow{SU} \:=\:0 \quad\Rightarrow\quad \langle -a,\,b,\,c\rangle\cdot \langle a,\,-b,\,c\rangle\:=\:0

    Therefore: . -a^2 - b^2 + c^2 \:=\:0 \quad\Rightarrow\quad a^2 + b^2 \:=\:c^2

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Rectangular parallelepiped vectors, etc
    Posted in the Geometry Forum
    Replies: 1
    Last Post: March 22nd 2011, 06:38 AM
  2. parallelepiped diagonal
    Posted in the Geometry Forum
    Replies: 1
    Last Post: December 5th 2010, 12:21 AM
  3. Parallelepiped
    Posted in the Geometry Forum
    Replies: 3
    Last Post: February 27th 2010, 01:08 PM
  4. problem (Rectangular Parallelepiped)
    Posted in the Geometry Forum
    Replies: 4
    Last Post: January 9th 2010, 05:27 AM
  5. parallelepiped
    Posted in the Geometry Forum
    Replies: 0
    Last Post: April 24th 2009, 07:30 PM

Search Tags


/mathhelpforum @mathhelpforum