Coordinate representation of tangent vector to the real projective plane. 
Let $a, b, c$ be three real numbers.
Let $X=\gamma'(0)$ be a field on the projective plane $P^2 (\mathbb{R})$, which has the usual maps:
$$
\phi_1 ([1 : x : y]) = (x, y), \qquad
\phi_2 ([x : 1 : y]) = (x, y), \qquad
\phi_3 ([x : y : 1]) = (x, y).
$$
and $$\gamma(t) = [(1 + ta)x : (1 + tb)y : (1 + tc)z], \qquad t \in \mathbb{R} .$$
Give the expression of $X$ in the first map.

I am trying to find $\phi_1(\gamma(t))$ to be able to compute $X$.
What I did:
I found that $\phi_1 (\gamma(t))=(\frac{(1+tb)y}{1+ta}, \frac{(1+tc)z}{1+ta})$. But is that valid given the difficulty (2)?
The difficulties:

*

*Why $X$ is defined as $\gamma'(0)$ as it depends not only on $\gamma$ but also on the maps?

*Why is $\varphi_1 ([1 : x : y]) = (x, y)$? I thought it was rather $\varphi_1 ([1 : y : z]) = (y, z)$. Maybe a notation abuse?

 A: (1) The vector $X = \gamma'(0) \in T_{[x, y, z]} P^2 (\Bbb R)$ does not depend on the choice of coordinate map $\varphi$, but its coordinate representation $\hat X \in T_{\varphi([x, y, z])} \Bbb R^2 \cong \Bbb R^2$ does.
(2) There seems to be some confusion here, stemming from the double-duty the variables $x, y, z$ are performing, namely as homogeneous coordinates on $P^2(\Bbb R)$ and as names for the arguments of the coordinate maps $\varphi_i$. Here, it's less misleading to write, e.g., $$\varphi_1([1, v, w]) := (v, w) .$$
Then, to write the coordinate representation $\hat \gamma$ of $\gamma$ with respect to the coordinate charge $\varphi_1$, we use homogeneity to write
$$[(1 + t a) x, (1 + t b) y, (1 + t c) z] = \left[1, \frac{(1 + t b) y}{(1 + t a) x}, \frac{(1 + t c) z}{(1 + t a) x}\right] .$$
So, taking $$v = \frac{(1 + t b) y}{(1 + t a) x} , \quad w = \frac{(1 + t c) z}{(1 + t a) x} ,$$
gives that the coordinate map $\hat \gamma := \varphi_1 \circ \gamma : \Bbb R \to \Bbb R^2$ is given by
$$\hat \gamma (t) = \varphi_1(\gamma(t)) = \varphi\left(\left[1, \frac{(1 + t b) y}{(1 + t a) x}, \frac{(1 + t c) z}{(1 + t a) x}\right]\right) = \left(\frac{(1 + t b) y}{(1 + t a) x}, \frac{(1 + t c) z}{(1 + t a) x}\right) ,$$
which agrees with your answer. To emphasize: In other coordinates, including in the affine coordinates defined by $\varphi_2$ or $\varphi_3$, the coordinate representation $\hat \gamma$ will generally be different.
