The definition of the tangent vector of a manifold In many textbooks of differential manifold, authors usually define the tangent vector on a manifold as below:
Definition: Suppose $m$ is a $n$-dimensional smooth manifold, $x\in M$, the tangent vector of smooth manifold $M$ at point $x$, if map $v: C_x^\infty\rightarrow\mathbb{R}$ is applied to all of the conditions below:
(1) $\forall f,g\in C_x^\infty, v\left(f+g\right)=v\left(f\right)+v\left(g\right)$;
(2) $\forall f\in C_x^\infty, \forall \lambda\in\mathbb{R}, v\left(\lambda f\right)=\lambda\cdot v\left(f\right)$;
(3) $\forall f,g\in C_x^\infty, v\left(f\cdot g\right)=f\left(x\right)\cdot v\left(g\right)+g\left(x\right)v\left(f\right).$
My questions are: 
(1)Why don’t we use the definition in the Euclidean space (treat a manifold as embedded in an Euclidean)? And why do we define a new one above? 
(2)What’s the idea or the purpose of giving the definition above? 
(3)How to treat the tangent vector in the Euclidean space as a particular case of the def. above?
Thanks of your attention to these questions and your opinions about them!
 A: (1) The definition of a manifold does not include an embedding into Euclidean space. It is not a priori clear that a manifold even can be embedded into some Euclidean space (though it is true, by the Whitney embedding theorem). Even though it can be done, it is convoluted, and there is no natural choice of embedding. There are actually many different embeddings, so if your definition of tangent vectors really depended on the embedding, you would run into trouble. And even if it didn't, it would be nice to have a definition which obviously doesn't depend on such things. 
(2) Because of (1), we want a definition of tangent vectors that does not depend on the manifold being embedded in some Euclidean space. The one you give is common, but there are others, e.g. equivalence classes of curves. It would be very desirable for these intrinsic definitions to line up with the definition we're used to when the manifold is embedded in Euclidean space, and this does happen in a nice way. For your definition, there is a natural isomorphism to the embedded-Euclidean definition by directional derivatives. 
(3) As in (2), you can think of these new tangent vectors as the directional derivatives in the directions of the old tangent vectors.
