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Thread: Vectors: parallelepiped

  1. #1
    Feb 2011

    Vectors: parallelepiped

    I need to show that the diagonals of a parallelepiped bisect each other. ie 2 lines going from opposite corner to opposite corner.

    ive tried experimenting with letting a b and c be vectors going in the 3 possible directions but i havnt got far. ive also tried to solve the problem by assuming the point M is in fact the bisect point and going from there but i keep hitting dead ends

    any help would be much appreciated
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  2. #2
    Senior Member abhishekkgp's Avatar
    Jan 2011
    it can be easily proved using euclidean geometry rather that vectors.
    consider two parallel planes plane 1 and plane 2. ABCD is a parallelogram in plane 1 and PQRS is a parallelogram in plane 2. these two parallelograms are such that AP || BQ || CR || DS.

    now ABCDPQRS is an arbitrary parallelopiped.
    to prove: AR bisects BS.
    observe that A,B,R,S are coplanar.
    also: AS=BR and AB=RS.
    so ABRS is a parallogram. since we know that diagonals of a parallelogram bisect each other we have AR bisects BS.
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