What exactly is Anthony Zee's definition of a vector in Einstein Gravity in a Nutshell? I am reading A. Zee's book Einstein Gravity in a Nutshell and encountered this definition of a vector on page 43:

An array of two numbers $p = \begin{pmatrix}
p^1 \\
p^2
\end{pmatrix} $ is a vector if it transforms according to $p' = R(\theta)p$

and then he proceeds to give an example that $\begin{pmatrix}
ap^1 \\
bp^2
\end{pmatrix}$ is not a vector if $a \neq b$.
I am utterly confused by this definition. Can anyone explain what does he mean in this case?
 A: Mr. Zee seems to be saying that, since we shall define the array $\vec{p}$ as a vector, we have
$$\begin{align}
p_1'&=R_{11}p_1+R_{12}p_2\\
p_2'&=R_{21}p_1+R_{22}p_2\\
\end{align}$$
Let us define a new array (not vector) $\vec q$ as
$$\vec q \equiv \begin{pmatrix} ap_1\\ bp_2 \end{pmatrix}$$
If $\vec q$ as an array transforms naively under the transformation $R$, we get
$$\begin{align}
q_1'&=ap_1'\\
&=R_{11}ap_1+\frac{a}{b}R_{12}bp_2\\
&=R_{11}q_1+\frac{a}{b}R_{12}q_2\\
q_2'&=bp_2'\\
&=\frac{b}{a}R_{21}ap_1+R_{22}bp_2\\
&=\frac{b}{a}R_{21}q_1+R_{22}q_2\\
\end{align}$$
On the other hand, if $\vec q$ is indeed a vector, it must transform under $R$ as simply
$$\begin{align}
q_1'&=R_{11}q_1+R_{12}q_2\\
q_2'&=R_{21}q_1+R_{22}q_2\\
\end{align}$$
If these two transformation rules — the "naive" and "true" ones — are to be the same, then we must have $a=b$. However if $\vec q$ is simply defined to be a vector from the start, and therefore transform under the second transformation rule, then there is no such restriction necessary.
Ultimately I do not see the point in this at all.
A: 
The components of a vector $\vec{p}$ are $x$ and $y$
$$\vec{p}=\begin{bmatrix}
  x \\
  y \\
\end{bmatrix}$$
where $x,y\in\Re$
thus the magnitude of $\vec{p}$ is $$|\vec{p}|=\sqrt{x^2+y^2}$$
if $\vec{p}\mapsto \vec{p}'$ the magnitude must be preserve: $$|\vec{p}'|=|\vec{p}|$$
with:
$$\vec{p}'=R\,\vec{p}\quad,\Rightarrow |\vec{p}'|=|\vec{p}^T\,R^T\,R\,\vec{p}|= |\vec{p}|$$
where R is a orthogonal transformation matrix .
so the magnitude preservation is fulfilled with any $x,y$ components!, this is also the case if  $x\mapsto a\,x$ and $y\mapsto b\,y$ where a and b are real numbers
