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# If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then verify that the points are the vertices of a parallelogram. - Mathematics and Statistics

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If A(3, 2, -1), B(-2, 2, -3), C(3, 5, -2), D(-2, 5, -4) then verify that the points are the vertices of a parallelogram.

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#### Solution

Let bar"a", bar"b", bar"c", bar"d" be the position vectors of A, B, C, D respectively w.r.t. the origin O.

Then bar"a" = 3hat"i" + 2hat"j" - hat"k", bar"b" = - 2hat"i" + 2hat"j" - 3hat"k", bar"c" = 3hat"i" + 5hat"j" - 2hat"k", bar"d" = - 2hat"i" + 5hat"j" - 4hat"k".

∴ bar"AB" = bar"b" - bar"a"

= (- 2hat"i" + 2hat"j" - 3hat"k") - (3hat"i" + 5hat"j" - hat"k")

= - 5hat"i" - 2hat"k"

∴ bar"DC" = bar"c" - bar"d"

= (3hat"i" + 5hat"j" - 2hat"k") - (- 2hat"i" + 5hat"j" - 4hat"k")

= 5hat"i" + 2hat"k"

= -(- 5hat"i" - 2hat"k")

∴ bar"DC" = - bar"AB"

∴ bar"DC" is scalar multiple of bar"AB"

∴ bar"DC" is parallel to bar"AB"

Also, |bar"DC"| = sqrt(5^2 + 2^2) = sqrt(25 + 4) = sqrt29

and |bar"AB"| = sqrt((-5)^2 + (-2)^2) = sqrt(25 + 4) = sqrt29

∴ |bar"DC"| = |bar"AB"|

∴ l(AB) = l(DC)

∴ opposite sides AB and DC of ABCD are parallel and equal.

∴ ABCD is a parallelogram.

Concept: Representation of Vector
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