Question 1067844
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Here's an easier way.

A(1,2,3) B(-1,-2,-1); C(2,3,2) and D(4,7,6)
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Vector AD = <4-1,7-2,6-3> = <3,5,3>
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Vector CD = <2-(-1),3-(-2),2-(-1)> = <3,5,3>
                                                       &#8594;    &#8594;
So sides AD and CD are both equal and parallel because AD = CD.
Therefore ABCD is a parallelogram because a quadrilateral with
a pair of parallel and equal sides is a parallelogram.

To prove it is not a rectangle, we find the dot product of two
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adjacent sides and show it is not 0. We dot AD which is <3,5,3> with
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CD = <4-2,7-3,6-2> = <2,4,4>

<3,5,3> • <2,4,4> = (3)(2)+(5)(4)+(3)(4) = 6+20+12 = 38 which
is not 0.  Therefore parallelogram ABCD is not a rectangle.

Edwin</pre>