Given 3 points (A, B, and C), express their centroid P in point+vector notation P = Q + sV.
I know the centroid of three points is
$$(A + B + C) \cdot (1/3)$$
but I'm unsure how to get that into something with a vector.
I'd appreciate any and all help.
Edit: I don't believe s has to be in the answer, because then it would be a line.
The trick is just to pick one of the points to be your starting point (basically your origin) and then represent each of the other points wrt that origin:
$$\begin{align}\frac 13A + \frac 13B + \frac 13C &= \frac 13A + \frac 13(A + (B-A)) + \frac 13(A+(C-A)) \\ &= A + \frac 13(B - A) + \frac 13(C-A) \\ &= A + \frac 13(\vec {AB} + \vec {AC})\end{align}$$
Once you get used to converting between barycentric and $P+\vec v$ forms this becomes second nature and you'll be able to do it in one step.