Express the feasible solution (3,1) as a convex combination of extreme points

Question

So i am given four extreme points:

$A: (0,0)$

$B: (8,0)$

$C: (2,3)$

$D: (0,1)$

and I need to express the feasible solution $(3,1)$ as a convex combination of extreme points. My professor did the following, he expressed it as:

$(3,1)=a(0,0)+b(8,0)+c(2,3)$

and solved for $a,b,c$

My question is the convex combination not unique and could I have done it as:

$(3,1)=a(0,0)+b(8,0)+c(0,1)$ instead?

Answer 1

Correct, it's not unique. Two non-collinear vectors in the plane form a basis.
Think of each point as its corresponding vector (from the origin to the point).

So for this representation to be unique, you need to have only 2 distinct points, e.g. A and B. And then all the convex combinations give you all points on the segment [A,B]. So right, you could have done it your way too.