Posted by Anonymous on Sunday, May 23, 2010 at 5:44pm.
Show that the quadrilateral with verticies at P(0,2,5), Q(1,6,2), R(7,4,2), and S(6,0,5) is a parallelogram.

CALCULUS  Damon, Sunday, May 23, 2010 at 6:47pm
vector PQ = 1 i + 4 j  3 k
vector SR = 1 i + 4 j  3 k
parallel
vector QR = 6 i  2 j + 0 k
Vector PS = 6 i  2 j + 0 k
parallel
done 
CALCULUS  drwls, Sunday, May 23, 2010 at 6:56pm
Your figure is embedded in three dimensional space, which makes it harder.
First consider the side lengths:
PQ: sqrt(1^2 + 4^2 + 3^2) = 5
QR: sqrt(6^2 + 2^2 + 0) = sqrt40
RS: sqrt(1^2 + 4^2 + 3^2) = 5
SP: sqrt(6^2 + 2^2 + 0) = sqrt40
Next look at the direction cosines. They are the same for the PQ and RS pair, and for the pair QR and SP.
Opposite sides are of equal length and parallel. Adjacent sides are connected. It must be a parallelogram. There must be a theorem for that. 
CALCULUS  Damon, Sunday, May 23, 2010 at 7:02pm
LOL , if opposite sides with four sides are parallel, it is a parallelogram.