# Vectors: parallelepiped

• Mar 3rd 2011, 02:16 AM
kumquat
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
• Mar 6th 2011, 04:58 AM
abhishekkgp
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.