Show that the construction of such a 3-dimensional shape is impossible in this case.

Question

Four points A(2; 0; 1); B(-1; 2; 3); C(3; 2; 2) and D(3,-6,-3) are expected to be the vertices of a 3-dimensional shape. Show that the construction of such a 3-dimensional shape is impossible in this case.

Answer 1

Can always make a plane in 3-space given three points

Using points A,B,C

find AB is vector (-3,2,2)

find AC is vector (1,2,1)

cross product provides a sized normal vector

N=2i-5j+8k

the dot product of N with (x-2,y-0,z-1)=0 provides plane ABC equation

2x-5y+8z=10

Point A was used, can use point B, or point C to find plane equation

notice that point D is on plane ABC

D=(3,-6,-3)

so A,B,C,D all on same plane

and no 3D shape possible.