Vectors transformation Give a necessary and sufficient condition ("if and only if") for when three vectors $a, b, c, \in \mathbb{R^2}$ can be transformed to unit length vectors by a single affine transformation.
This is  just a bonus question that the teacher gave us but i dont know even know whats it talking about or how to go about it 

Assume that $a,b,c$ are different. (If not, they will lie on a circle and that circle could be affine transformed to the unit circle around the origo.)

Prop.: $a,b,c\in\Bbb R^2$ are not collinear iff there is an affine transformation $\phi(x)=Mx+v$ such that $\phi(a),\phi(b),\phi(c)$ are unit vectors.

$\Rightarrow$: If $a,b,c$ points on plane are not collinear, then there is a circle that contains them, let's call its center $q$, and its radius $\rho$. Then the affine transformation $\ x\mapsto x-q\ $ followed by $\ x\mapsto x/\rho$ will map each $a,b,c$ to the unit circle.
$\Leftarrow$: If $a,b,c$ are collinear, then they will stay collinear under any affine transformation, hence they are not all going to fit the unit circle for sure, as a line can meet a circle at most in $2$ points.
