# Linear independence of three points

What is the best way to prove if three points are linear dependent?

This is my current way to do it:

$$\mathbf{v_a} = \mathbf{b}-\mathbf{a} \\ \mathbf{v_b} = \mathbf{c}-\mathbf{a} \\ \mathbf{v_c} = \begin{pmatrix} -\mathbf{v_{a,y}} \\ \mathbf{v_{a,x}} \end{pmatrix} \\ \phi = \mathbf{\hat{v}_c} \cdot \mathbf{\hat{v}_b} \\ \phi < 10^{-3}$$

$\phi$ is the $arccos$ of the real angle between the two vectors. This works good if the polygon or the three points of the polygon have well defined distances. But what if the polygon looks like this:

The distance between $P_0$ and $P_1$ is much bigger than between $P_1$ and $P_2$. $P_0$, $P_1$ and $P_2$ are of clearly not linear dependent.

I'm searching for a scale-independent algorithm for that.

Do you have any ideas or a reference to a paper I can read about it?

Normalize $v_a$ and $v_b$ and your algorithm should no longer depend on the length of the edges, unless you consider numerical stability.