Let $v(x,y)=(x,y)$ be a vector field. Plottet, it looks like this:
So this is a very simple question: Why do the arrows get bigger in length the further away they are? It is clear to me, that if we take some random $v=(x_1,y_1)$ it's length describes how far it is away from the origin. Is that what the length of the arrows in the plot should indicate?
A vector field $V$ on $\mathbb R^2$ maps a point p to tangent vectors $V($p$)$. It's not necessary that this tangent vector has greater magnitude with p further from the origin. Let $V:(x,y)\rightarrow(1,1)$ and you will have a constant vector field that gives you arrow of same magnitude everywhere and pointing in the same direction.