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Question

By vector method show that the quadrilateral with vertices $A (1, 2, - 1), B (8, - 3, - 4), C (5, - 4, 1), D (- 2, 1, 4)$ is a parallelogram.

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Solution

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As we know that diagonals of parallelogram bisect each other.
In the figure, $O$ is midpoint of diagonals $A C$ and $A D$ then

$(\frac{1 + 5}{2}, \frac{2 - 4}{2}, \frac{- 1 + 1}{2}) = (\frac{8 - 2}{2}, \frac{- 3 + 1}{2}, \frac{- 4 + 4}{2})$ [Finding coordinates of $O$ ]

$(3, - 1, 0) = (3, - 1, 0) .$

Hence, $A B C D$ is a parallelogram.

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