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.

#### 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.