Normalizing directional derivatives

Question

Let we are in a flat 2D space, and we have 3 vector fields: $u = x e_y$, $v = e_x + 2 e_y$, $w = 2 e_x + 4 e_y$. Now if we calculate directional derivative of u in the direction of v we get $\partial_x (x e_y) + 2 \partial _y(0e_y) = e_y$. However, directional derivative in the direction of w gets us $2e_y$. Why are the directional derivatives different? w is just an scaled version of v, the direction is the same. Shouldn’t the results be normalised?

Answer 1

You are saying that for any two vector fields $X$ and $Y$ and a non zero constant $c$ we should have $XY=X(cY)$? That's clearly incorrect since the vectors $e_{x_i}$, are derivations, which are linear.