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

• 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?

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Shortest notation as in? –  Inceptio Mar 11 '13 at 8:46
@Inceptio: How do you usually denote each of those questions above? Can you make it shorter? –  kiss my armpit Mar 11 '13 at 8:48
I always just see $\operatorname{Proj}_{\vec{B}}(\vec{A})$. –  Alexander Gruber Mar 13 '13 at 22:54
@AlexanderGruber: and the complementary one? –  kiss my armpit Mar 13 '13 at 22:55
@GarbageCollector I don't know of a notation for that one. –  Alexander Gruber Mar 13 '13 at 22:57

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.

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Agreed 100% except that I think your last notations should be e.g. $\bf{A}^\perp_\bf{B}$; it's the component of * A * normal to the axis defined by B, rather than vice versa (and I might go with $\bf{A}_{\perp\bf{B}}$, though that comes across a little more awkward). –  Steven Stadnicki Mar 14 '13 at 17:17
@StevenStadnicki Yes! You are right, that is much better. –  Alexander Gruber Mar 14 '13 at 19:08

It is a $pr_{\vec{B}}(\vec{A})$ for the first case. I think, there is no notation for the second.

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