Dot product, inner product, and projections

[nomadreid]

TL;DR Summary

Letting u,v be unit vectors, the length of the projection of u onto v is u dot v, whereas the inner product <u|v> is the projection of v onto u. Why the difference?

In simple Euclidean space: From trig, we have , for u and v separated by angle Θ, the length of the projection of u onto v is |u|cosΘ; then from one definition of the dot product Θ=arcos(|u|⋅|v|/(u⋅v)); putting them together, I get the length of the projection of u onto v is u⋅v/|v|.
Then I read that the inner product <u|v> is the result of the projection of v onto u.
Of course one could just say that the dot product is commutative, but the reverse order of what is projecting onto what seems a bit odd.
Either: where is my mistake, or: What am I missing?
Thanks in advance.

 

[FactChecker]

There are a ton of different sign and notation differences in math and physics. The best that you can hope for is that any given book or article is consistent. Even that is sometimes violated and a book/article notation convention may be dependent on the context.

 

[fresh_42]

If you are interested in the topic you might want to read
https://arxiv.org/pdf/1205.5935.pdf

It is mathematics, but well written and a nice overview.

 

[DrClaude]

TL;DR Summary: Letting u,v be unit vectors, the length of the projection of u onto v is u dot v, whereas the inner product <u|v> is the projection of v onto u.

That's a question of semantics. For me, $\mathbf{u} \cdot \mathbf{v}$ is projection of $\mathbf{v}$ on $\mathbf{u}$, not the other way around. For real-valued vectors, there is no difference because of commutativity. For complex-valued vectors, it matters because the two inner products are complex conjugate of each other,
$$
\braket{u | v} = \overline{\braket{v | u}}
$$
Note tat another common notation for an inner product is $(u,v)$, for which the convention is most often that $v$ is the quantity that will be complex-conjugated.

 

[nomadreid]

Thanks, FactChecker, fresh_42 and DrClaude.
fresh_42: The book looks very clearly laid out, and I have downloaded it, as it will certainly be helpful.

DrClaude: I believe you have a typo in your note that the two inner products are complex conjugates of one another: there should be a line over one of the pair, or an asterisk, or however one chooses to indicate the complex conjugate.

 

[DrClaude]

DrClaude: I believe you have a typo in your note that the two inner products are complex conjugates of one another: there should be a line over one of the pair, or an asterisk, or however one chooses to indicate the complex conjugate.

There is an overline. Maybe it is a question of MathJax rendering. What I see is

 

[Mark44]

There is an overline. Maybe it is a question of MathJax rendering. What I see is
View attachment 333411

I see the same, both here and in your previous post.

 

[FactChecker]

I see the same, both here and in your previous post.

On my Windows 10 PC Firefox browser, I don't see that in the post, only in the .png image.
On the Chrome browser, I see it correctly in the post.
On my Samsung Android tablet Chrome browser, I see it correctly in the post.