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.

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