the duality between quaternions and vectors
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- Опубліковано 11 вер 2024
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The way my math teacher talked about quaternions is as followed : a quaternion a + b.i + c.j + d.k has two parts : the scalar part a (same role as the real part for complex number) and a vector part b.i + c.j + d.k (same role as the imaginary part for complex number). If you consider two pure vectors (a = 0), their product gives a quaternion whose scalar part corresponds to the scalar product and the vector part to the cross product ("vectorial" product in french).
That's exactly how quaternions have been used in physics in the 19th century, e. g. for formulating Maxwell's equations.
"Clifford algebra is a right there man"
Another way to understand (unit) quaternions is that they use a 720° degree system. They are _literally equivalent_ to axis-half angle notation. Notice how the 720° maps to 360°. :-)
It may also help that the imaginary unit _i_ represents a +90° rotation around the Z axis. That makes it easier to understand that the imaginary components _i, j, k_ represents the axis of rotation for the (unit) quaternion.
@@dang-x3n0t1ct Bivectors say hello. :-)
Someday GA (Geometric Algebra) / CA (Clifford Algebra) will be taught in high school.
Everyone who knows geometric algebra expected something like that. But you don't need nether quaternions nor cross product. Multivectors completely cover all the use cases.
This was the translation between Hamiltonian (and Tate) physics by quaternions, and Heaviside and Maxwell (and others) physics by vectors.
Максвелл первоначально использовал кватернионы, и Хевисайд использовал обозначения кватернионов...
@@ruby_linarisI treatise on electricity and magnetism (maxwell) and Heaviside, for instance Electromagnetic Theory, vectors are used. In fact Heaviside has a whole section on why vectors (notation used is modern boldface) are superior to quaterianans.
@@Calcprof no, first version Maxwell wrote on Quaternions, vectors crate later... second edition wrote on vectors. yes, Heaviside start vector-war, so His was noname into phisics.
Geometric Algebra demystified quaternions for me, as well as complex numbers and other interesting systems.
Geometric Algebra is nonsense
@@michaelaristidou2605 It isn't, though. It's highly effective, and it originated from the same time period as the foundations of vector analysis.Even this channel has videos on Clifford Algebras. Much of particle physics uses these methods for spinors analysis.
@@michaelaristidou2605 Is calculus nonsense? The world revolves around calculus despite it being an informal use of analysis. Meanwhile, geometric algebra is useful for unifying most algebraic objects in mathematical physics despite it being an informal use of Clifford algebra. A century from now, people will likely use geometric algebra the same way as calculus.
@@mrervinnemeth "As a result GA’s relationship to mainstream mathematics is tenuous. It is considered a kooky, crackpotty sideshow. Even worse, as a result of being dubious and un-self-aware, the movement ends up losing everyone except for a particular type of… zealous person… who write about it with a sort of pseudooreligious zeal, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism. When in fact it is just not very compelling at the moment."
@@spaghetti1383 "As a result GA’s relationship to mainstream mathematics is tenuous. It is considered a kooky, crackpotty sideshow. Even worse, as a result of being dubious and un-self-aware, the movement ends up losing everyone except for a particular type of… zealous person… who write about it with a sort of pseudooreligious zeal, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism. When in fact it is just not very compelling at the moment."
Reminds me a lot like Clifford Algebra
it's is the same thing but more generalized
Quaternions are exactly isomorphic to G_{0,2}(R), which is the 2d geometric algebra with basis vectors with magnitude squared equaling -1
@@mrervinnemeth G(0,2) has 1 scalar (1), 2 vectors (i,j) and 1 bivector (ij), and if we associate ij to k, we get ij=-ji=k, ik=iij=-j=jii=-iji=-ki, and jk=jij=-ijj=i=-kj, which is the multiplicative structure of the quaternions, and the additive structure is also the same as the quaternions, where any element can be represented as a+bi+cj+dk.
@@mrervinnemethI am aware of the even subalgebra of G(3) being isomorphic to the quaternions, however the algebra G(0,2) is also isomorphic to the quaternions. In G(0,2) we have to use the entire algebra for the isomorphism, as in G(0,2) we have 3 non-scalar components, which become i,j,k (you can select any order for them as it works either way). The isomorphism is G(0,2)H: ii, jj, ịjk, 11.
@@mrervinnemeth the even subalgebra of G(2) is indeed the complex numbers, however G(0,1) is also the complex numbers, where the full algebra is the complex numbers. By G(n,m) I am meaning having n basis vectors squaring to 1, and m basis vectors squaring to -1.
I like this turn towards more geometric topics in Penn's recent videos
The associativity of the Quaternions is one of the miracles of Maths
Quaternions might be treated as elements of R x V^3, where V^3 is a 3-D real vector space. They then multiply, thusly:
q.r = ( α, *a* ) x ( β, *b* ) = ( αβ - (*a*.*b*), α*b* + β*a* - *a*x*b* )
Then, reversing Penn's procedure, we use the vector identity:
*a* x (*b* x *c*) - (*a* x *b*) x *c* = (*b*.*c*).*a* - (*a*.*b*).*c*
To prove associativity; i.e. that q.(r.s) - (q.r).s = 0
which is fun to do (if you like symbol-bashing)
Could do a follow up video of this now that you demonstrated how quaternions and vector operations are related through the properties of the dot and cross products. With that we could now transition from this to show how quaternions are also related to the cosine through the dot product as well as the formula cos(t) + i*sin(t). From there we can even go one step further and couple this with the Fourier Series and transforms, especially the fast Fourier transform and its inverse. As an added bonus we could even throw in a few RK methods to solve for a few basic ODEs. RK = Runge Kuta and ODE being simply an Ordinary Differential Equation solver.
I am intrigued with things related to Fourier Analysis. However, before we dive in, it should be established why Quaternions are a preferred method of performing rotations in 3D compared to using conventional Euler Angles along with the caveats to Euler Angles while using the commonly known rotation matrices.
All of these techniques are very useful in 3D Graphics Rendering, Physics Simulations, Animations, and Audio Processing applications as well as many other fields. Try creating a model of a weather pattern related to hurricanes or tornadoes. Try decoding a foreign complex wave like signal such as an unknown source of sound, light, radiation, or anything else that has a frequency. Try determining the amount of energy released during an explosion in a complex dynamic system. These are just a few uses of them.
These are some of my favorite fields within mathematics and to bring them together into a single functional physics problem, it doesn't get much better than that. Now that's a true form of integration! What else can we derive from them?
16:09
In the set of quaternions there are infinitely many square roots of -1 (as, indeed, of any negative real number): any linear combination ai+bj+ck with a^2+b^2+c^2=1 is a root of -1.
Octonians obey alternative property: (x x) y = x ( x y ), ( y x ) x = y ( x x ) for all octonians x and y. Makes me wonder if there is an R^7 O duality possible like for associative property of quaternians and R^3 H . Great video! Love the series. Cheers.
7:01 🤯
It makes me wonder what a non-zero "real" part of the quaternion would correspond to in vector land. I'd love to see a followup if possible.
It could be used in Minkovskji space, for instance one can re-frame maxwel'sl equations. The question is : does it worth?
Maxwell originally wrote his equations in terms of quaternions. The familiar form using vector analysis notation was produced by Oliver Heaviside.
Does it make any sense to add torque and energy and get a quaternion?
Ayo , you already did about this topic .... I wish this video were released 1 month ago.
Hi, can you try the 2024 Benelux Math Olympiad Problem 2? It's quite an interesting problem
Yesss I love more abstract math like this!
Does someone know any advanced graphing calculator that can take complex plain and not only the real one
Would be nice to see an application
👀
I noticed that a lot of clueless physicists gathered around here, claiming that "yea all that is geometric algebra and blah blah blaaah".
Ok, sorry to break it to you but the stuff in the video are real math where geometric algebra is pure nonsense.
"As a result GA’s relationship to mainstream mathematics is tenuous. It is considered a kooky, crackpotty sideshow. Even worse, as a result of being dubious and un-self-aware, the movement ends up losing everyone except for a particular type of… zealous person… who write about it with a sort of pseudooreligious zeal, as if the only reason GA is not mainstream is that they are being oppressed by close-minded traditionalism. When in fact it is just not very compelling at the moment."
I wish I could see the heighty algebra in all this but all I see (perceive?) at cross product as shown @ 0:01 is ... :
hold x values fixed (that is (twisting?) contribution on i components?) and contribution on those points is given by values of y's and z's
Then similarly for fixed y's (note change in sign. Hmm determinants?) and fixed z's
Then for nice permutations of i, j, k ij=k jk=i ki=j has symmetry of some sort whereas other orderings (not nice) introduce skew-symmetry e.g. kj=-i and so forth
I suppose 𝐢𝐢=i(jk)=(ij)k=𝐤𝐤=k(ij)=(ki)j= 𝐣𝐣 so by deduction 𝐢𝐢= 𝐣𝐣=𝐤𝐤 and these equal -1 by given
That is not to say I understand what is being presented in tangible sense but what I perceive in a behavioral sense.
And in this case the behavioral sense seems to make physical properties manifest and that is a wonder of its own. It is almost philosophical to ponder why it should be like that?