Talk:Properties of Dot Product

Just a note:-

(I think) the "Alternative proofs" are circular: the fact that $u \cdot v = \| u\| \|v\| \cos\theta$ is assumed, but in the proof of this fact, $(4)$ and $(5)$ are used.

Of course it works because of the first set of proofs, but perhaps it's worth a note to this effect --Linus44 08:59, 11 May 2011 (CDT)

The idea behind the alternate proofs is actually to show that if you take $u \cdot v = \| u\| \|v\| \cos\theta$ as the definition of the dot product, the results are still valid. I put a note at the top of the Alternative Proofs section trying to explain that, but I'm afraid I was rather asleep at the time. I'll take another look tomorrow, and possibly rework the Definition:Dot Product page and move Cosine Formula for Dot Product to Equivalence of Dot Product Definitions or some such.

Ah right, point taken. --Linus44 05:55, 12 May 2011 (CDT)

Side note: I think I've made progress on the alternative proof for (4) in the general case. Essentially, if you examine the projections of $\vec u$, $\vec v$, $\vec u + \vec v$, and $\vec w$ into the plane formed by the X-axis and each other axis in turn, you can show by equating components through the angle addition formulas that the vectors must be equal in each of the projections. It would then follow (non-trivially I think :( ) that the vectors are equal outside of the projection. Messy bugger, this. --Alec (talk) 02:02, 12 May 2011 (CDT)

Surely it can't be proved from the cosine form in the general case ($\R^n$, $n > 3$), since in this case $u \cdot v = \| u\| \|v\| \cos\theta$ is the only available definition of the angle between to vectors? --Linus44 05:55, 12 May 2011 (CDT)

The messiness probably stems from the fact that the conventional "vector" in Euclidean real space (of however-many dimensions) is only a specific instance of the general concept of "vector", and while you can prove equivalence between the two definitions in this particular context, it does not necessarily translate into definitions of vectors on spaces where "cosine" is not defined.

Wherever you've got an "iff" you can specify the definition of an entity in terms of either side of that iff, but it doesn't half get confusing working out what depends on what. We have the same trouble with sine and cosine themselves. I wonder what the best approach to this is - maybe set up a completely parallel thread of separate, but heavily cross-linked, pages.

I would also advise that when bundling up a whole load of results into one page, we put each result on its own page and then create a master page of transclusions, like with Subset Equivalences and the more ambitious Trigonometric Identities and the similar approach on Binomial Coefficients. --prime mover 02:37, 12 May 2011 (CDT)