Vectors in non-orthogonal systems In a non-orthogonal coordinate system, what is the physically significant difference between the components of a vector on the skew axes and its projection onto each axis? Why would one want to find one over the other?
 A: For simplicity let's work in 2D, and take as our axes two unit vectors $\hat{e}_i$ and $\hat{e}_j$. We'll consider some vector $\hat{F}$:

In our coordinates we can write the vector as $(F_i, F_j)$, where $F_i$ and $F_j$ are just numbers. The vector $\hat{F}$ is then expressed as the vector sum:
$$ \hat{F} = F_i \hat{e}_i + F_j \hat{e}_j $$
I've drawn the two vectors $F_i \hat{e}_i$ and $F_j \hat{e}_j$ in red.
But there is no physical meaning to the numbers $F_i$ and $F_j$. They are just numbers that depend on whatever basis vectors we choose. They can't be observables because changing our basis vectors doesn't change the physical system we're observing but it does change $F_i$ and $F_j$.
On the other hand the dot products $\hat{F}\cdot\hat{e}_i$ and $\hat{F}\cdot\hat{e}_j$ are physical observables because they give the force we would measure in the directions of $\hat{e}_i$ and $\hat{e}_j$ respectively. The reason why the dot products have physical significance is because now $\hat{e}_i$ and $\hat{e}_j$ correspond to something physical i.e. they are the directions in which we measure the forces.
