SOLUTION: show that points (1,2,3);(-1,-2,-1);(2,3,2) and (4,7,6) are vertices of parallelogram but it is not a rectangle.

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Question 1067844: show that points (1,2,3);(-1,-2,-1);(2,3,2) and (4,7,6) are vertices of parallelogram but it is not a rectangle.
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
A%281%2C2%2C3%29
B%28-1%2C-2%2C-1%29
C%282%2C3%2C2%29
D%284%2C7%2C6%29
The vector from B to A is
.
The vector from C to D is the same:
,
So AB and DC area parallel,
going in the same direction,
and with the same length,
That makes ABCD a parallelogram.
To be a rectangle the diagonals AC and BD should have the same length.
However, their length squared are different:
and

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!


Here's an easier way.

A(1,2,3) B(-1,-2,-1); C(2,3,2) and D(4,7,6)

       →
Vector AD = <4-1,7-2,6-3> = <3,5,3>
       →
Vector CD = <2-(-1),3-(-2),2-(-1)> = <3,5,3>
                                                       →    →
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
                                            →    
adjacent sides and show it is not 0. We dot AD which is <3,5,3> with
→
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