Deriving the Law of Cosines With Geometric Algebra

In this short, I show a quick derivation of the law of cosines using geometric algebra.

I thought I would try YouTube shorts out, so I'm planning on making one a week. This only took me several hours to make, so it shouldn't interfere with my main videos too much. Look forward to seeing more of these in the future!

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[caspermadlener]

Gonna be honest, this was pretty sick.
I wonder whether the cosine rule for quadrilaterals can also be derived this way.

[evandrofilipe]

That's actually insanely cool. I wonder for which other polygons you get cool expressions.

[sounakroy]

I have been doing this without knowing it's geometric algebra.

[danielfrt]

A connection between the fundamental theorem calculus, sequences and trigonometry. This should be the general derivation of the law of cosines.

[curiousaboutscience]

Some of these comments are interesting. Fun stuff!

[artkalbphd]

Isn't a.b = ab cos(theta) normally derived from the properties of the dot product using the law of cosines? This seems like a circular argument.

[ayokay]

I'm just here for the pretty colors and squiggly lines

[TheDigiWorld]

I hate schools and colleges for not showing us such stuff

[anuraagkumar]

Holy fuck that’s so fucking smart oh my god

[GarnetGoldGunner]

Wow that’s really helpful in my everyday life

[JonasLampert-sw]

What is meant with the "square of a vector"? Or should it be the absolute value?

[fredeisele]

given n vectors forming a loop in n dimension euclidean space:
0 = Si (Ai^2) + 2(Sij (Ai.Aj))
Ai : one of the vectors
Si : the sum of the terms

e.g. n = 4
0 = a + b + c + d
squared, gives
0 = a^2 + b^2 + c^2 + 2[a.b + a.c + a.d + b.c + b.d + c.d]
cool

[I_S_H_A_A_N]

huh I am in class 9 I learnt this but this looks harder that rocket science

[xy]

While this is interesting, it is overly complicated. You can get the exact same result by doing the same process with the dot product instead of the geometric product. Nonetheless, I am sure the geometric product will work better in similar scenarios.

[Dayanto]

Took me a moment to figure out what was happening when you skipped the step where the bivectors cancel. But I guess it makes sense to ignore them since they aren't relevant to the problem.

[Abc-osvm]

Why didn't you say it is al kashi theorem if it were Pythagoras theorem you would be shouting

[joefuentes]

Geometric Algebra is the absolute shit