Definition talk:Dot Product/Real Euclidean Space
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This definition for the dot product should be restricted to only Euclidean space. In a generic normed vector space, it not only fails to be equivalent to Definition 1, but it fails to be an inner product.
First, observe:
| \(\ds c^2\) | \(=\) | \(\ds a^2 + b^2 - 2 a b \cos \theta\) | Law of Cosines | |||||||||||
| \(\ds 2 a b \cos \theta\) | \(=\) | \(\ds a^2 + b^2 - c^2\) | ||||||||||||
| \(\ds a b \cos \theta\) | \(=\) | \(\ds \frac {a^2 + b^2 - c^2} 2\) | ||||||||||||
| \(\ds \vec a \cdot \vec b\) | \(=\) | \(\ds \frac {\norm {\vec a}^2 + \norm {\vec b}^2 - \norm {\vec a - \vec b}^2} 2\) | Definition of Dot Product/Definition 2 |
Now, consider $\R^2$ with the taxicab norm:
| \(\ds \paren {\tuple {0, 1} + \tuple {1, 0} } \cdot \tuple {0, 1}\) | \(=\) | \(\ds \tuple {1, 1} \cdot \tuple {0, 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\norm {\tuple {1, 1} }^2 + \norm {\tuple {0, 1} }^2 - \norm {\tuple {1, 0} }^2} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {4 + 1 - 1} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2\) |
But:
| \(\ds \tuple {0, 1} \cdot \tuple {0, 1} + \tuple {1, 0} \cdot \tuple {0, 1}\) | \(=\) | \(\ds \frac {\norm {\tuple {0, 1} }^2 + \norm {\tuple {0, 1} }^2 - \norm {\tuple {0, 0} }^2} 2 + \frac {\norm {\tuple {1, 0} }^2 + \norm {\tuple {0, 1} }^2 - \norm {\tuple {1, -1} }^2} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + 1 - 0} 2 + \frac {1 + 1 - 4} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
This violates Axiom $2$.
I was considering suggesting that the definition of angle between vectors be modified as well, but there may be an interesting geometric interpretation of that, so I'll leave it be. --CircuitCraft (talk) 19:10, 27 April 2023 (UTC)
- Clearly then the two definitions cannot be considered as equivalent, but that definition $1$ is more general than $2$ which is a special case of $1$.
- All too much for me at the moment. I think I need to bow out. --prime mover (talk) 22:10, 27 April 2023 (UTC)
- Renamed as appropriate. I am in the process of restructuring this area. Be aware it may be some time before the scaffolding comes down. --prime mover (talk) 12:51, 10 May 2023 (UTC)
- May do another refactoring in due course: to rename "General Context" version the actual full definition of "Dot Product", making sure to establish what the domains of the most general scalar and vector are in the context of the top-level abstract algebraical definition of Vector Space, because it is too easy to gloss over the limitation by taking the usual context of real Euclidean space for granted. Every single result and proof in this category needs to be checked for the highest level of abstraction possible, with appropriate analysis of exactly where the limitations are. --prime mover (talk) 15:01, 10 May 2023 (UTC)