Vector notation I'm working with vectors and resolving them into their components on vector form, like this:
$$\vec{F} = F_1\hat{\imath}+F_2\hat{\jmath}.$$
I'm wondering whether we need to put vector notation over the components on the RHS, or whether we don't put vector notation on the components?
 A: 
The $\hat{\imath}$ and $\hat{\jmath}$ are already vectors. (That is, they are $\vec{\imath}$ and $\vec{\jmath}$, the hat $\hat{}$ symbol just indicates they are of unit length.)
In the image above the vector $\vec{r} = 3 \hat{\imath} + 2.5 \hat{\jmath}$ is literally described in terms of a vectorial sum of scaled basis vectors $\hat{\imath}$ & $\hat{\jmath}$.
The expression
$$\vec{r} = 3 \hat{\imath} + 2.5 \hat{\jmath}|$$
literally means: take $\hat{\imath}$, scale it to 3 times its size; then take $\hat{\jmath}$, scale it 2.5 times its size; then add them together using vector addition, to get the vector $\vec{r}$.

Which is why 3 & 2.5 are scalars. They scale things. (The etymology of the word is a historically a little deeper than that, but that is the gist of it.)
So you don't put the $\vec{}$ arrow on them, they are just numbers — the vector arrow is already on $\hat{\imath}$ and $\hat{\jmath}$ (although it's masquerading as a hat).
Similarly, in
$$\vec{F} = F_1 \hat{\imath} + F_2 \hat{\jmath},$$
$F_1$ & $F_2$ are just numbers (scalars).
Now, you may sometimes reuse the labels to represent component-vectors; e.g. you could say:
$$\vec{F}_1 = F_1 \hat{\imath},$$
$$\vec{F}_2 = F_2 \hat{\jmath},$$
and
$$\vec{F} = \vec{F}_1 + \vec{F}_2,$$
but note that now $\hat{\imath}$ and $\hat{\jmath}$ have "disappeared" because they are included in $\vec{F}_1$ and $\vec{F}_2$, respectively.
So $\vec{F}_1$ and $F_1$ are different. One is a vector, an arrow of a certain size (magnitude), pointing in some direction. The other is just a number telling you how much bigger (or smaller) $\vec{F}_1$ is compared to $\hat{\imath}$ (including negative values, when the direction is flipped).
$$\vec F_1 \neq F_1$$
$$\vec F_2 \neq F_2$$
A: The $F_1$ and $F_2$ are the magnitude of the components and don't need to be vectors.  The $\hat{\imath}$ and $\hat{\jmath}$ show the direction of the components.
A: 
look at this figure
$$\vec F=\vec F_x+\vec F_y=F_x\,\vec{e}_x+F_y\,\vec{e}_y$$
where $~F_x~,F_y~$ are the  components  along the x and y axis and
$ ~\vec{e}_x~,\vec{e}_y~$ are basis  vectors; $~\vec{e}_x\cdot\vec{e}_x=1~,\vec{e}_y\cdot\vec{e}_y=1~,\vec{e}_y\cdot\vec{e}_x=0$
A: $F_1$ and $F_2$ are scalars, so should be left as you've shown them. $\hat{\imath}$ and $\hat{\jmath}$ are unit vectors. The 'hat' (^) symbol indicates both their vector nature and their unit magnitude, so no additional vector indication is needed.
A: You don't need to put vector notation on RHS until you're writing $\hat{\imath}$ or $\hat{\jmath}$ symbol. If you don't write these symbols, then you need to put the vector notation.
A: Another standard notation is to put an underline to indicate vectors (when writing on paper, this is often replaced with bold face in typeset documents).
I think that notation works better for your example than the notation with the arrow over the top. On paper I would indicate unit vectors with an underline and a hat, so your formula would look like
$$\underline{F} = F_1 \underline{\hat{\imath}} + F_2 \underline{\hat{\jmath}}.$$
I think this is more clear because the notation is consistent;  vector quantities always have a line underneath them, and there is no exception for unit vectors.
A: Gosh, there are so many different notations out there. Just use what works for you, as there is no cannonical form here.
Some examples are
$$\begin{array}{r|l}
\text{Notation} & \text{Expression}\\
\hline \text{Vector} & \vec{F}=F_{1}\hat{i}+F_{2}\hat{j}\\
\text{Overline} & \overline{F}=F_{1}\overline{\imath}+F_{2}\overline{\jmath}\\
\text{Underline} & \underline{F}=F_{1}\underline{\hat{\imath}}+F_{2}\underline{\hat{\jmath}}\\
\text{Boldface}^* & \boldsymbol{F}=F_{1}\boldsymbol{\hat{\imath}}+F_{2}\boldsymbol{\hat{\jmath}}\\
\text{Column Vector} & \boldsymbol{F}=\begin{pmatrix}F_{1}\\
F_{2}
\end{pmatrix}\\
\text{Row Vector} & \boldsymbol{F}=\begin{pmatrix}F_{1} & F_{2}\end{pmatrix}^{\top}\\
\text{Over Right Arrow} & \overrightarrow{F}=F_{1}\overrightarrow{\imath}+F_{2}\overrightarrow{\jmath} \\
\text{Basis Vectors}*&\boldsymbol{F}=F_{1}\boldsymbol{e}_{1}+F_{2}\boldsymbol{e}_{2}
\end{array}$$

*

*(*) The boldface notation, using \boldsymbol{F} is the most common on professional papers, with either unit vectors or basis vectors.


*I personally use slanted variables for scalars ($F$), bold slanted for vectors ($\boldsymbol{F}$), and upright bold for matrix values ($\mathbf{F}$). Some special matrix values such as the identity matrix, I use just upright values ($\mathrm{F}$) since they stand out more.
In all of them, the scalar components do not have any decoration or modifications. So the magnitudes $F_1$ and $F_2$ are always as is.
Also, note that for unit vectors using $\hat{i}$ or $\hat{\imath}$ is common but use what works for you, as long as you are consistent and it is obvious what your intent is. To remove the dot over the i and j use \imath and \jmath.
A: In 3 dimensional space, a vector $\vec{F}$ can be represented as a linier combination of unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$ that are perpendicular each another.
$$
\vec{F}= F_x\, \hat{x}+F_y \,\hat{y}+F_z\, \hat{z}
$$
where $F_x$, $F_y$, and $F_z$ are just numbers (or scalars).
