Parametrization of a Parallelogram: Mapping Rectangles onto Planar Regions

[za3raan]

Homework Statement

(a) Show that the four points r1 = (1, 0, 1), r2 = (4, 3, 5), r3 = (6, 4, 6) and
r4 = (3, 1, 2) are coplanar and the vertices of a parallelogram. Let S
be the closed planar region given by the interior and boundary of this
parallelogram. An arbitrary point of S can be written as the convex linear
combination

$\sum a_{j}r_{j}$ for j= 1 to j=4 $\sum a_{j}=1$ $0<a_{j}<1$

Show that the vertices, edges and interior of the rectangle R = [0, 1]×[0, 1]
are mapped onto the vertices, edges and interior of S by the linear map
(parametrization) r = r(u, v) : R → S

r = (x, y, z) = (1 + 3u + 2v, 3u + v, 1 + 4u + v), (u, v) ∈ [0, 1] × [0, 1]

Homework Equations

Not sure

The Attempt at a Solution

I've shown that the four points are coplanar and the vertices of a parallelogram, however I really have no idea about the rest. Some guidance would be very much appreciated!

 

[za3raan]

Sorry! I just saw that we can't use attachments to present question. I've edited the above post with the full question.