Components of Vectors My questions have to do with vectors and scalars in general.  However, for ease of communication, I'll craft my questions around the following vector equation:
$$\overrightarrow{r}=x\widehat{i}+y\widehat{j}+z\widehat{k}$$ 
As I understand it, x, y, and z constitute components.  Are components signed numbers (a.k.a. scalars) or magnitudes (a.k.a. absolute values)?  If the former, what exactly do the signs on these numbers mean?  If the latter, where do the signs on components come from and what exactly do they mean?
 A: The components are real numbers, and as such they can have either sign. If, for example, $x$ is negative, it means that $\vec r$ points in the direction opposite to $\hat{i}$. You can see this for yourself: work in two dimensions for simplicity (so $\vec r = x \hat i + y \hat j$), and make up a bunch of vectors with different values of $x$ and $y$, including positive, negative, and zero. See what happens.
A: You are really asking for the difference between the magnitude and the component of a vector.  
The magnitude is always a positive quantity whereas the component can be both negative and positive.
For ease suppose that in one dimension $\vec r=-3 \hat i$
You can interpret this in one of two ways.
$\vec r=(-3) \hat i$ where $-3$ is the component of $\vec r$ in the $\hat i$ direction.
$\vec r=+3 \hat{(-i)}$ where $+3$ is the magnitude of $\vec r$ in the $\hat{(-i)} $ direction where $\hat{(-i)} $ is a unit vector in its own right which happens to be in the opposite direction to unit vector $\vec i$ which you could call $\hat I$.
Note that $\hat I =-1 \hat i$ where unit vector $\hat I$ has a component of $-1$ in the $\hat i$ direction.
You can think of it as once you have chosen a fixed set of reference axes (directions) then then you will use components of the vector in those fixed reference directions to define the vector.  
If you choose the magnitude then you will choose the appropriate unit vector to define the vector.  
