SOLUTION: Show that the points A(1,0,2), B(3,-1,3), C(4,2,1) and D(2,3,0) are the vertices of a parallelogram.

Question 1057010: Show that the points A(1,0,2), B(3,-1,3), C(4,2,1) and D(2,3,0) are the vertices of a parallelogram.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Show that the points A(1,0,2), B(3,-1,3), C(4,2,1) and D(2,3,0) are the vertices of a parallelogram.
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Find the side lengths:
AB = sqrt(1 + 1 + 4) = sqrt(6)
BC = sqrt(14)
CD = sqrt(6)
AD = sqrt(14)
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Show they're in the same plane:
If the cross product of AB & AD is the same vector as the cross product of BC & CD the points are coplanar and it's a parallelogram.
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AB = 2i - j + k
BC = i + 3j - 2k
ABxBC = -i + 5j + 7k
---
BC = i + 3j - 2k
CD = 2i - j + k
BCxCD = i - 5j - 7k
Only the directions are different (180 degs different) --> the 4 points are coplanar.
CDxBC = ABxBC --> coplanar
--> it's a parallelogram.
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There might be a simpler, shorter test for this.
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