Is there the shortest notation for a vector obtained by projecting $\vec{A}$ onto $\vec{B}$? 
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*Is there the shortest notation defined for a vector obtained by projecting
$\vec{A}$ onto $\vec{B}$?

*Is there the shortest notation defined for the complementary vector of a vector obtained by projecting  $\vec{A}$ onto $\vec{B}$?


Is it ok if I used $\vec{A}\parallel \vec{B}$ for the first case and $\vec{A}\perp\vec{B}$ for the last one?
 A: Actually, after thinking about it some more, I now remember seeing the following used in my abstract linear algebra class:
$$\operatorname{proj}_{\vec{B}}(\vec{A})$$
$$\operatorname{perp}_{\vec{B}}(\vec{A})$$
and these in my 2nd year physics curriculum:
$$\operatorname{Proj}_{\parallel}(\vec{B},\vec{A})$$
$$\operatorname{Proj}_{\perp}(\vec{B},\vec{A})$$
I am not sure if there is a standard notation for these.  Personally, I like the first one, since the second does not make immediately clear what is being projected onto what.
I don't think that using $\vec{A}\parallel\vec{B}$ and $\vec{A}\perp \vec{B}$ because these are conventionally used as statements about $\vec{A}$ and $\vec{B}$, i.e. $\vec{A}\perp\vec{B}$ should be read "$\vec{A}$ is orthogonal to $\vec{B}$" rather than an expression which means "the component of $\vec{B}$ orthogonal to $\vec{A}$."  I think if you wrote, for example, $$B\perp \operatorname{perp}_{\vec{B}}(\vec{A})$$ that any reader would recognize this as a true statement (perhaps a tautology).
If you're writing in a situation where you can define your own notation, and you really want something short, I might suggest $\vec{A}_{\vec{B}}^\parallel$ and $\vec{A}_{\vec{B}}^\perp$ or (my personal preference) $\mathbf{A}_{\mathbf{B}}^{\parallel}$ and $\mathbf{A}_{\mathbf{B}}^{\perp}$.  $$\mathbf{A}=\mathbf{A}_{\mathbf{B}}^{\parallel}+\mathbf{A}_{\mathbf{B }}^{\perp}$$ just looks slick.
A: It is a $pr_{\vec{B}}(\vec{A})$ for the first case. I think, there is no notation for the second.
