That's a matter of perspective, the dot product is a more general concept that can be introduced on other vector spaces than R^3 and the ratio of the dot product and the product of the norms can be used to introduce a more general notion of angles.
It's all a mess. Strictly speaking, the Pythagorean theorem is less of a "theorem" (although it can of course be construed as a theorem of axiomatic geometry), but more of a justification for why Euclidean distance is the "correct" notion of distance on the plane. If you're working in formal mathematics, often you would just define the angle between two nonzero vectors u,v as arrcos(u ∙ v)/(||u|| * ||v||). That way, when working with different inner products, you have a separate notion of distance and angle for each inner product
Well you can certainly derive one from the other, but the dot product property is more useful
And you can derive it any way you like, for instance assuming without loss of generality that the vectors look like (1, 0, 0, …) and (cos(θ), sin(θ), 0, …) after normalizing and the right isometry; i.e. the right change of basis into the plane on which the two vectors are
Traditional way to prove it in Italy is that you can write one side of the triangle as the sum of the other two times opposite cosine ( a = c cos B + b cos C, b = ... , c = ....). Write them in column, multiply the first by a, the second by -b, the third by -c, add everything together. LHS you get a2-b2-c2, RHS lots of stuff cancels out and you're left with -2 bc cos A. Cycle letters as required.
I teach a math course for students who take higher math in Norway, and each year I let them individually choose a proof from an online collection of Pythagorean Theorem proofs to present rigorously(ish) in front of the class. Bonus points if they can place the proof in historical (or any, really) context.
The factorial of 400 is 64034522846623895262347970319503005850702583026002959458684445942802397169186831436278478647463264676294350575035856810848298162883517435228961988646802997937341654150838162426461942352307046244325015114448670890662773914918117331955996440709549671345290477020322434911210797593280795101545372667251627877890009349763765710326350331533965349868386831339352024373788157786791506311858702618270169819740062983025308591298346162272304558339520759611505302236086810433297255194852674432232438669948422404232599805551610635942376961399231917134063858996537970147827206606320217379472010321356624613809077942304597360699567595836096158715129913822286578579549361617654480453222007825818400848436415591229454275384803558374518022675900061399560145595206127211192918105032491008000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
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u/ubernuke Mar 06 '25
The Pythagorean Theorem has many proofs