Show that the points A(1, 2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram. Show that ABCD is not a rectangle.

To prove: Points A, B, C, D form parallelogram.

Formula:

The distance between two points (x₁,y₁,z₁) and (x₂,y₂,z₂) is given by

D=

Here,

(x₁,y₁,z₁)= (1, 2, 3)

(x₂,y₂,z₂)= (-1, -2, -1)

(x₃,y₃,z₃)= (2, 3, 2)

(x₄,y₄,z₄)= (4, 7, 6)

Length AB =

=

=

=

=

Length BC =

=

=

=

=

Length CD =

=

=

=

=

Length AD =

=

=

=

=

Length AC =

=

=

=

=

Length BD =

=

=

=

=

Here, AB = CD which are opposite sides of polygon.

BC = AD which are opposite sides of polygon.

Also the diagonals AC and BD are not equal in length.

Hence, the polygon is not a rectangle.