We know that the basis vectors $\{\textbf{e}_r,\textbf{e}_{\theta},\textbf{e}_z\}$ for cylindrical coordinates and the basis vectors $\{\textbf{e}_x,\textbf{e}_y,\textbf{e}_z\}$ for Catesian coordinates at a common point in space are related by a rotation described by the following equalities $$\begin{cases}\textbf{e}_r=\cos\theta\textbf{e}_x+\sin\theta\textbf{e}_y\\\textbf{e}_{\theta}=-\sin\theta\textbf{e}_x+\cos\theta\textbf{e}_y\\\textbf{e}_z=\textbf{e}_z\end{cases}$$
Let $\textbf{v}=R\textbf{e}_r+\Theta\textbf{e}_{\theta}+Z\textbf{e}_z\in \mathbb{R}^3$, then from above relationships we have $$\textbf{v}=R(\cos\Theta\textbf{e}_x+\sin\Theta\textbf{e}_y)+\Theta(-\sin\Theta\textbf{e}_x+\cos\Theta\textbf{e}_y)+Z\textbf{e}_z\\=(R\cos\Theta-\Theta\sin\Theta)\textbf{e}_x+(R\sin\Theta+\Theta\cos\Theta)\textbf{e}_y+Z\textbf{e}_z$$
However, we know from coordinates conversion that $\textbf{v}=R\cos\Theta \textbf{e}_x+R\sin\Theta \textbf{e}_y+Z\textbf{e}_z$, which is different from above expression (except the $z-$ component). Why is this so?
This seemingly "inconsistency" between coordinates conversion and basis conversion is also refelcted by dot product computation: $\textbf{v}\cdot\textbf{v}=R^2+\Theta^2+Z^2$ under cylindrical coordinates $\{\textbf{e}_r,\textbf{e}_{\theta},\textbf{e}_z\}$, but it is clearly not true in Cartesian coordinates because the legnth of $\textbf{v}$ is in fact $\sqrt{r^2+z^2}$. Does this mean that the dot product (or length of a vector) derived under the cylindrical coordinates are different from the ones under Cartesian coordinates?
Could anyone point out the subtlety behind above "inconsistency" for me?
Best Answer
Dot product (of coordinate vectors) is indeed dependent on coordinates. For instance, the vector
$$ e_1 $$ has coordinate vector $v = (1,0,0)$ in the basis $e_1, e_2, e_3$ which has $$v \cdot v = 1,$$ but the same vector, expressed in the basis $$\frac{1}{2} e_1, e_2, e_3$$ has coordinate vector $$ w =(2, 0, 0), $$ and $w\cdot w = 4$.
post-comment additions:
To explain the other problem you're having:
When you write "Let $$ \textbf{v}=R\textbf{e}_r+\Theta\textbf{e}_{\theta}+Z\textbf{e}_z\in \mathbb{R}^3 \ldots,$$
you've already gone off the rails. Because the basis you've written at the top of your post is the basis for the space of "vectors based at the point whose coordinates, in the polar system, are $(r, \theta, z)$." It's not a basis for $\mathbb R^3$, because in the context of $\mathbb R^3$, the values $r$ and $\theta$ aren't even defined!
This'll become obvious once I make things concrete: Let's the the vector $v$ where $R = 2, \Theta = \frac{\pi}{2}, Z = 1$. In the expression for $e_r$ that you plugged in, what's the value of $\theta$? You seem to have plugged in $\Theta$, but why? For the basis to make sense, you need the $(r, \theta, z)$ coordinates of the point at which you're using it as a basis, but you don't have those...so you've used the nearest thing, typographically, as a substitute. There's really no justification for that.
Some Gratuitous Advice
A question for you: When you replaced the lower-case $\theta$ and $r$ with their upper-case versions, did something in the back of your mind say, "Hey, wait a minute...these are actually different!"? And did you then perhaps say "Yeah, but they're the only "r" and "theta" I can see in the formulas I've got, so I guess I have to use them!"? Because that voice in your head was the warning that you were doing something wrong, and needed a deeper understanding before proceeding.
I spend a good deal of time programming, and I find debugging about 10 times as hard as programming. That's pretty much true for math as well, and listening to that little voice is part of the way to avoid debugging (in both contexts).