Ok, this is my favourite #SoME2 as it is in the spirit of the original 3blue1brown videos - taking an obscure memorised algebraic fact and turning it into a statement of visual obviousness. Kudos.
We actually never learned the formulas for cos(x + y) or sin(x + y) in normal maths class. I'm now in my last year of hs, and I chose to participate in an optional maths course, where we learned about these formulas using complex numbers (10x more elegant imhu). If you didn't already know, the proof goes as follows: (Note that I don't have the greek letter theta, so I'll use x) e^(i(x + y)) = e^(ix + iy) = e^(ix) * e^(iy) = (cos(x) + isin(x))*(cos(y) + isin(y)) = cos(x)cos(y) + cos(x)*i*sin(y) + isin(x)cos(y) + isin(x) * isin(y) = ( cos(x)cos(y) - sin(x)sin(y) ) + i( cos(x)sin(y) + sin(x)cos(y) ) (Split into real and imaginary parts) As such, we have the formula for cos(x + y) in the first term, and the formula for sin(x + y) in the second by identification, since e^(i(x + y)) = cos(x + y) + isin(x + y)
This proof is equivalent to the one I gave in the video, with the only difference that it presupposes some nontrivial machinery for it to actually work out. The main reason why I avoid this approach is not because it's invalid or difficult to explain once you know the properties of the complex exponential function. Rather, the difficulty lies in explaining why the complex exponential function works the way it does. For instance, can you give me a valid explanation for why e^(ix) should equal cos(x)+isin(x)? And furthermore, how do you prove that this definition gives us an exponential function that satisfies the equation e^(ix+iy)=e^(ix)*e^(iy) for all x and y (without already knowing the addition formulas for sine and cosine, that is)? These questions are not necessarily easy to answer, and I would expect most people to struggle with them quite a bit, even with some explanation given. This is, in part, why I chose to prove the statement the way I did. One of the "downsides" of my approach is that I need to spend some time to lay the groundwork explaining what 2d vectors are and how rotations of them work, but once I have that the proof is basically just one line. The upside of my approach, however, is that all the preprequisite linear algebra that I need to explain it is elementary, and it's simple to explain intuitively with the use of graphical aids.
Personally, I like using complex numbers to derive the angle sum formulae because they follow directly from the dead simple power sum formula: e^(a+b) = (e^a)(e^b). Using Euler's formula e^ix = cos(x) + isin(x), you can then do e^i(a+b) = (e^ia)(e^ib) = (cos(a) + isin(a))(cos(b) + isin(b)) = cos(a)cos(b) + icos(a)sin(b) + isin(a)cos(b) - sin(a)sin(b). Separate the real and imaginary parts and you get sin(a+b) = cos(a)sin(b) + sin(a)cos(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b). This also follows the geometric interpretation used in the video since multiplying by a complex number is scaling by the magnitude (which is 1 in this case) and rotating by the angle, so you're really just rotating another rotation. Regardless of how you do it, the truth will pretty much always be that sine and cosine belong together and it's so much easier to handle them together rather than separately. sine and cosine are just two halves of a whole, the whole being one of the simplest operations in geometry: rotation.
Yeah, multiplication of complex numbers is a great way to derive these formulas. The only slight problem I have with this method is that you kind of already have to understand why e^(a+b)=(e^a)*(e^b) still holds when a and b are complex numbers. Many times that identity is proved by alredy knowing that the addition formulas for sine and cosine hold, which leaves us with a bit of a conundrum. But in case that I already know that the formulas are true, then I think that using this method is probably the easiest way to derive the formulas if one should forget them. And I totally agree when you say that sine and cosine belong together. They are really just two sides of the same coin.
@@TheArmchairIntellectual I thought that the power law was a part of the definition when extending to non-natural number exponents, complex or otherwise, with e^x specifically having the extra requirement of d/dx e^x = e^x. With the latter, it's possible to get an idea for the complex exponential even without using euler's formula. d/dx e^ix = ie^ix via chain rule. i itself is a rotation by 90°, so the derivative of e^ix is always perpendicular to the function itself. The main area I've seen that property is in rotational dynamics, where the speed of an object remains constant, but the velocity gets accelerated in a perpendicular direction toward some center.
@@angeldude101 As far as I'm aware the exponential function is usually defined using its power series representation so that it can be extended to general Banach algebras, and then properties are derived from the series expansion. But yeah, there is no real issue having the power rule as a part of the definition of the exponential function, and then basically just use a differential equation to seal the deal; the necessary boundary condition e^0=1 seems to follow automatically from this meaning that it becomes redundant in a sense... (one can of course use the differential equation + a boundary condition as well, but then the power law has to be proven). I've actually thought a bit about how to use the power rule without using differential calculus to define the complex exponential, and I have some ideas of maybe making a video about it in the future if I can think of a good way to present it in the way I want to. In either case, no matter which way we use to define the complex exponential there is some explaining that needs to be done before it's obvious that e^{ix}=cos x + i sin x (using varying degrees of differential calculus), and I kind of felt like no matter which explanation I used it wouldn't be possible to showcase the main thing I wanted to in this video; after all, the idea I wanted to showcase was the use of a dictionary to solve a problem in an unexpected way, which is an approach to problem-solving that I just find super appealing.
its fun until we realise there is Circular reasoning. Euler's formula requires taylor series taylor series requires differentiating sin, cos, exp diff sin requires evaluating limit (sin(x+h)-sin x)/h , h->0 that limit requires sum angle formula which requires euler's formula :(
cool vid. i thought you'd encode rotations as matrices and use that to rotate the vectors but i was pleasantly surprised to see that you thought about rotations more abstractly and intuited what useful algebraic properties it would have and how the angle sum formulas would follow. i would've been happy to see this when i was first learning about the compound angle formulas because it feels nicely motivated and like an obvious consequence of some properties about rotations unlike the standard way it's taught.
I completely agree. The actual reason I came up with this proof about two years ago was precisely because of the pedagogical potential I saw in it. I was taking a teaching course at the time, and it really got me thinking about a few things, among them this video idea.
@@TheArmchairIntellectual It feels like you did the matrices without doing the matrices! Personally, although I think not bringing matrices into it was by far and away the right call, it might’ve been worth flashing up a brief “for the interested” on screen just showing the proof in matrix multiplication form. If it goes over your head, fine; if either you know matrices or can figure it out, then excellent. Still a great video, and made my jaw drop.
@@krozjr5009 Yeah, after all linear maps and matrices are the same thing, so we can definitely say that I talked about matrices without mentioning them. I just felt like it would take too much effort to explain them, without really making the core points of the video any clearer in the end, so I decided that things would work out just fine if I didn't even mention them :)
Hmm I didn't learn that 'ugly' proof, I learned a different one where you draw right triangles inside a rectangle. Granted it only works for acute angles but it's at least more motivated and less symbol-shuffley than that one. But even so, the vector proof is so much nicer! Great video!
I've never seen that proof but yeah, at this moment it seems to me like we will lose a lot if we don't use rotations to justify the addition formulas, no matter how we try to avoid them. And this is probably not very strange considering how closely tied sine and cosine are to rotations.
Yeah I definitely also learned some geometric mumbo jumbo. Spent a lot of days trying to make enough intuitive sense of the steps to be able to reconstruct it on my foggy shower wall.
This is how this was taught in Poland 2007-2010. I remember that class: loads of chalk, plowing through identities and final arrival at concise and satysfying formula. Fun times. It absolutely can be entertaining.
This felt like an expansion of the Essence of Linear Algebra series, it was so good! As an educational video creator myself, I understand how much effort must have been put into this. Liked and subscribed, always enjoy supporting fellow small creators :)
That's the first time I saw sin and cos as Cartesian coordinates of a unit circle! Back in trig and pre-calc I asked my teacher what was the formula inside the sin function , both teachers just said to look it up on the table or use a calculator, I never got any understanding from that☹️
That's unfortunate really. Viewing sine and cosine as the coordinates of the unit circle is by far the most common way to geometrically describe them for arbitrary angles, and it really helps in understanding why they are so useful when considering rotations.
@@TheArmchairIntellectualThis is the first time I've seen in and I saw math courses from multiple universities not describing it that way... Probably because they expect teacher to teach basics, but they did fuck all. After some reading on sinus and cosinus I just learned that I only knew the definition for triangles... I didn't know the definition for circles even existed. WTF now I can go back 2y and finally understand some of the lectures I just could not comprehend....
The art of math is to see the bigger picture of any mathematical structure one is trying to construct. The way you started with the definition of a general vector (to have operations that can create new elements using pre existing elements, not mentioning the exact wording but essence remains the same) and then developed necessary motivation needed to create a geometric vector, correlate algebraic vectors to geometric counterpart so that the correlation is one to one (aka unique) was breathtaking. My mind went boom and said eureka (though instead of finding elegant solution of some problem it was due to coming across this video). Excellently explained , ur voice was a cherry on the topping. Hope u go a long way. And don't stop making videos. Want to relish more of these mathematical delicacies.
Thanks for explaining everything thoroughly, even if a lot of those things are obvious to math enthusiasts. A lot of people just skip over the explanation and take stuff for granted.. That's missing the whole point of these videos!
Yeah, it's all about the intended audience. And here I didn't want people unfamiliar with trig and linear algebra to feel completely left out, so I decided to include stuff that some people might find a bit superfluous.
My favorite way of proving those trig identities is using complex exponentials. It's so easy and simple, I don't even memorize the relations, just work with complex exponential whenever I need them
Of course, this is a very simple way to do it once we fully understand how the complex exponential works (and how it is connected to rotations in the complex plane). My only problem with this kind of proof is that Euler's formula e^{ix}=cos x+i sin x is normally just stated rather than fully explained when first introduced, and the fundamental properties of the complex exponential tend to be proven using the addition formulas for sine and cosine. So unless we already have an independent understanding of the complex exponential and how it connects to rotations, we end up with a sort of circular argument.
We'll see how it goes, no matter what happens next I'm just super happy both with what I managed to make and that people seem to actually appreciate it. If nothing else, the entire experience surrounding this video has been a huge confidence boost :)
the proof i learned (well, found and remembered) was to construct line segments with angle theta and (theta+phi) over a unit circle, then construct lines to form the right triangle from the endpoint of the latter to the former, and from that intersection point to the x axis. extend the latter line vertically, and construct another line to form a third right triangle with the first point mentioned. essentially, we have constructed a right trapezoid with the diagonal line having length 1, sectioned into 3 right triangles by 2 lines intersecting at a right angle somewhere along the far edge. that edge, in fact, has the length sin(theta+phi), but it is split into two sections with lengths we can calculate: the right triangle from theta to (theta+phi) has hypotenuse 1 and feet of lengths cos(phi) and sin(phi). those feet are the hypotenuses of the other 2 right triangles. those other 2 right triangles must be similar to each other because, if the first one has angles a, 90, and b, we know that a+b=90 by the triangle inner angle sum, and we know that b+90+n=180, where n is the first angle of the second right triangle in question. therefore, n must be a, and the other angle must be b. we know that a is theta because it is the origin angle of the right triangle defined by the x axis and the line with angle theta from it. therefore the full far edge, sin(theta+phi), is equal to the sum of its two partitions, which, as is clear from the construction, have lengths cos(phi)sin(theta) and sin(phi)cos(theta). this proof is surprisingly difficult to put to words alone, but it is extremely simple visually, especially if you take it for granted that the outer two of the right triangles are similar. you literally just draw the right trapezoid sectioned into the three right triangles, and the angle sum formula falls out.
Do you perhaps have a link to this proof, where it's accompanied by a visual explanation of the line constructions involved? I think that I'm kind of messing it up somewhere when I try to construct the lines you describe... Anyway, thanks for providing a different way to prove these formulas, I really like it when there are multiple different ways to prove the same thing :)
@@TheArmchairIntellectual i made a reply, but it seems to have been automatically deleted, likely for containing a link. in place of the link, here's instructions to the same end: open up a desmos calculator tab and append "/wpaptamvod" (without the quotes of course) after /calculator. it should contain the construction, together with some notes and two sliders to adjust the angles (limited to acute positive angles, as that range is where the identity being proven is most clearly shown by the construction--and I didn't want to keep having to think about the bounds of the equations that would be needed so that the lines would display properly for all cases with obtuse angles) one thing that i just noticed: t and p are expressed in radians on the sliders, but i wrote the angles in the notes in degrees. what i mean should be fairly clear despite this.
Nice video. Your method using vector is really nice. My preferred method is using complex numbers and Euler's formula, which reframes the problem into one of purely algebra.
Yeah, I think that using Euler's formula is maybe the easiest way in practice once you understand why Euler's formula actually works (but I didn't want to explain that in this video :)), since it doesn't require us to build a formula for rotations from scratch. So in a pinch, if you forget what the formulas are then using that method is really convenient. In a way though, that proof is very similar in spirit to the one I gave since the core of both proofs really is exploiting rotations in the plane (whether we view it as complex or not).
Excellent video. Just a minor comment I want to make and I hope it doesn't come off as snarky but I would like to add my 2 cents. I wouldn't really call the trig identity for sum of angles or ANY trig identity for that matter a "proof" but rather a derivation. They exist and are defined as such because of so. If someone doesn't believe a trig identity, then you could "prove" it to them by showing them a logical derivation which you have done with perfection. Good stuff. Make more.
I like the idea that a proof is only really useful if it's comprised of obvious steps. Of course there are proofs that can't be simplified to fit this category - and that's the thing, these obscure proofs feel more like axioms where you either know the result or don't, without really being able to prove it yourself. I think the best kind of proof is the one you can fall back on in case you forgot the relation in hand. In the best case you wouldn't even have to learn the result. Like here, with the addition and multiplication identities of the trig functions, it's way easier to do the 5 minute derivation, with linear algebra for example, than to recite these seemingly meaningless formulas.
I wouldn't go so far as to say that a proof is only useful if it's comprised entirely of obvious steps, rather that if we have a proof that isn't then there is probably something about the statement that we have yet to understand fully. I mean, most of the time mathematicians will initially find a proof of a theorem that is really complicated and ad hoc, but that proof usually serves as a starting point for actually dealing with the statement with more confidence. In either case, I think there is a lot more to mathematical proofs which often gets overshadowed by the statements that they're meant to prove.
29:39 Is it just a coincidence that the x-coordinate of the rotation of the vector [a, b] by an angle theta is precisely the derivative with respect to theta of the y-coordinate of the same rotated vector? I'm gonna have to look into this on my own... Oh and great video btw :) I'm taking linear algebra right now so this is really helpful and interesting.
Hmm, I haven't really thought about it but there is probably some way to explain why that is the case. Maybe has something to do with the complex exponential function, but I don't know at the moment.
I guess it might be related to something on the lines of this If we imagine an object rotating along a unit circle at the rate of 1 revolution per unit time, we note these facts: 1. The velocity is just the rate of change of its position (i.e. its derivative). Hence, *v* = d *r* /dt 2. The velocity vector is perpendicular to its position vectors at any given moment. Hence *v* ⊥ *r* 3. The lengths of position vector and velocity vector must be equal as | *r* | = | *v* | = 1 When object's position was *x* , its velocity was just *y* . Rotating this snapshot by θ just corresponds to Rot( *x* ) and Rot( *y* ). We already noted that *v* = d *r* /dt. So *v* = Rot( *y* ) is just d/dt (Rot( *x* )). So I guess this somehow shows y component being the derivative of x component. I guess this is a very crude way to see it 😅. But only this much is coming to my mind at the moment. Note: I wasn't able to find the cap symbol, so through *x* and *y* I mean x hat and y hat (the unit vectors)
Wouln't the argument you make in minute 3 just valid for differences under 90 degrees? Also in the final argument, how do you extend the statement that the vector (0,1) going to (-sin(),cos()) to greater angles?
Hi there! I'm not quite sure which exact argument you're referring to at the 3 minute mark. If it's the argument that |PQ|=|AB|, then this argument really doesn't depend on the size of the angle difference at all, as it is only concerned with lenght segments that are rotated around the origin. Feel free to comment if I misunderstood which argument you referred to here. As for your second question, this is something that is easiest done by just checking manually for the four different nontrivial cases (angle between 0 and 90 degrees, between 90 and 180 degrees, between 180 and 270 degrees, and between 270 and 360 degrees); after this everything works on account of periodicity. This can be done in each case by making basically the exact same argument that I gave in the video for each quadrant, and checking that the signs really are correct in each case. One could conceivably try to find a more elegant argument here to deal with general angles without checking different cases, but I think that would be a waste of time tbh.
While this is certainly possible, it's trickier than it seems at first glance if one wants to avoid circular logic. Basically, one has to motivate why e^(ix)=cos(x)+i sin(x) and then independently motivate why e^(ix+iy)=e^(ix)e^(iy) (of course without using the trig identities that we wish to prove). At the end of the day I only see two general approaches one could take. Firstly, we have the geometric approach where one argues using rotations in a manner similar to what I did in this video. Secondly we have the differential approach, where one could work using Maclaurin/Taylor series expansions or differential equations. However, this second approach requires some calculus know-how before one could hope to pull it off, and it really seems like overkill to me when trying to prove some trig identities.
Hmmm, I have to say that I didn't intend for what is happening at this time to be considered as "awesome", but I guess I'm glad someone enjoys this part as well...
In general, row and column vectors are dual to one another, so a row vector in this context would correspond to a 1-form. I think 3blue1brown talked about this in his EssenceOfLinearAlgebra series in connection with the dot product. My opinion is that the dual of a linear space can be a bit difficult to motivate from an intuitive point of view unless one is already familiar with linear algebra to begin with, so this is part of the reason why I chose not to cover this nuance in the video.
I guess that depends a bit on exactly what you find most troublesome to understand. I tried to make the video accessible to anyone who is familiar with high school mathematics (like some trigonometry, a decent grasp on the algebra of real numbers), but of course that can be a bit difficult for me to do perfectly since it was such a long while since I was in high school. Anyway, I think it would be easier for me to answer your question if you could point to something specific in the video that you found difficult to follow along with.
Hi, If Cartesian Coordinate System gives just scalar multiplication to basis vector (i, j). What are polar coordinates doing... In which one in just scalar distance(r) and the other angle is?.. Which type of tensor is angle in the set of (distance, angle, area volume). Could you make sense of this....
I'm not quite sure what it is that you are asking to be perfectly honest. As far as I'm aware, polar coordinates are not commonly used when talking about linear algebra, but if you really wish to connect "polar coordinates" (where we don't have the zero element well-represented) with some sort of "tensor" I suppose your best bet may be to consider the [angle] part of the polar coordinate be a rotation of the plane (i.e., it's an element of the group of linear transformations of the plane called SO(2)), and to consider the [distance] part to denote a scaling operator, which scales vectors in the plane by a factor [distance].
@@kartikeyedunite Of course, you could make this definition. But then you would lose out on the regular properties of polar coordinates that one typically associates with them. Basically, with this definition you would end up with polar coordinates working like regular cartesian coordinates from an algebraic perspective, and it would have nothing to do with the standard way we actually add complex numbers. In other words, it usually doesn't work to try and have your cake and eat it too.
Sure am! In fact, this is the real difference between a novice and an expert in almost any topic, not just mathematics; the expert has just looked hard enough at enough things to know how to make things easier for themselves. While a bit of a simplification, I think it's good to keep this in mind, since so many people tend to overestimate the importance of talent and ignore the amount of hard work that goes into mastering anything.
nice proof! I can't help but ask; since you said vectors are "things that we can add up and scale to get the same things" don't rotations also check out? You gave the addition and scaling rules in the video. Im a 4th year CS student, so you can assume some basic knowledge :)
We usually think of rotations in the plane as forming a structure called a group, with the group operation being composition. Since they are linear maps, they can be identified with a set of matrices, often denoted by SO(2,R); this is a subgroup of GL(2,R) often called the special orthogonal group. Then rotations can be viewed as matrices and the group operation becomes matrix multiplication. However, I haven't really heard this group described as a vector space, since it's not clear how one is supposed to scale a rotation and get back a rotation afterwards; since the group is abelian we may view the group operation as "addition" if we want to, but the issue with scaling by real constant is something I don't know how one would resolve.
Not on this channel as of yet (and I'm quite frankly not entirely sure what a 3D version of "this" would be). I have some projects I'd like to cover on this channel in the future (although it will have to wait until after I finish my PhD), we'll see if I find something interesting to talk about in 3D linear algebra. I'm not sure what topic that would be at the moment, but who knows what the future holds?
Sorry, javascript burn my brain. I mean vector in 3D or N dimensions. How it rotate, and translate in 3D space. you did a great job! thank you thank you
@@lkda01 I see. If we discuss rotations in higher dimensions this is usually generalized to groups of linear transformations called the special orthogonal (or unitary in the case of complex numbers) groups, which is an interesting topic to discuss. When it comes to translations I guess it may be best to discuss the matter in terms of affine transformations, which are transformations that are "almost" linear. While I'm no expert on affine transformations it may be interesting to go over sometime in the future.
I used the python library called ManimCE (the community edition of Manim that was created by Grant Sanderson). It took some time to animate, but I think the end result was worth the effort.
i am not math oriented.....as a college student i found the trig functions to be unpleasant and easy to forget or confuse.....it was like learning a new language, but i found studying russian and french were more congenial and less mechanical than mastering logarithms..........................in a way its sad that so much of the important industry depends on higher math because probably 97% of people find math to be intimidating and unpleasant, and no matter how many problems you solve and master there is always another problem ahead that is worse than the problem you just laboriously wrestled with for 2 hours.
how do you square a distance? how do you square an orange? how do you square an angle? how do you square a '57 chevy? how do you square a trombone? how do you square a dog? i understand how you square a number ...... a distance from miami to atlanta is not similar to the number 12 ...... you can fly an airplane to atlanta, you cannot fly an airplane to the number 12 ....... 12 is an idea in the mind ..... distance is a physical separation of two locations
This proof still doesn't make it intuitive, because that isn't what proofs do. Like, no crap that rotating leads to that formula, but the specific form of that formula itself is not made intuitive depending on the proof you do, and you still have to work the proof to see what form the angle summation formula(s) take(s) If that form was intuitive, I would be able to write the form just by looking cos( a + b ) and knowing intuitively that it equals cos(a)cos(b) - sin(a)sin(b).The moment you do any work, it isn't intuition anymore (however, knowing exactly how to proceed to arrive at the solution or proof form is also a form of intuition, but that means the two proofs you showed are not better than each other in any respect; in fact, intuition allows one to proceed with either proof in the correct manner and on the first try)
I'm not sure how to respond to this comment in the best way. While I agree that it's never going to become intuitively obvious that the formula for cosine is exactly what it is, if we understand rotations as linear transformations it is pretty intuitively clear that there should be a formula for the sine and cosine of a+b using the sines and cosines of a and b. So while it's not going to be completely intuitively obvious that cos(a+b)=cos a cos b- sin a sin b, it can make perfect intuitive sense that the formula must look very similar to this, and then finding its exact form is more of a formality than a challenge. And I'm not entirely sure what you mean that proofs "do", exactly. There are many proofs out there that are both hard to follow and will not make it clear at all why something works on a conceptual level, while there are other proofs that basically explain why the statement they are supposed to prove make perfect sense. While the basic requirement for a proof is that it is a logical derivation of a certain fact, this is not the only thing a proof can be good for, and there is a lot of difference in a "good" and a "bad" proof, even when both proofs are perfectly logically valid. At the end of the day, while I do have some disagreements with you I do really appreciate that you took the time to explain your perspective (it really helps me to see why someone might not view my video in the same high regard that I do; while I'm trying to be objective about it it's always going to be hard not to be somewhat biased...).
@@TheArmchairIntellectual I mostly agree with everything you just said. Except when I think of 'intuitive' I think I can immediately understand it and know the form with my intuition alone. If I have to do any work at all, then that isn't intuition anymore. There can be intuition in knowing how to proceed such that the work that you do leads to the right answer on the first try, or in the right direction without much fuss, such as knowing to use rotations to lead to an answer proof of angle addition formula. But I still cannot say that the formula itself is intuitive; just that rotations makes doing the proof intuitive. Also, you say I may not view your video in the same high regard, but I greatly enjoyed this video and think that it is of top quality I dual majored in physics and mathematics, so I don't know if my conceptions are different, but a proof is a logical set of deductions that establishes a truth I am of the view that all of mathematics is just one big tautology. Or at least the stuff that is consistent is one big tautology; whether it is complete or not is irrelevant I've never really used proofs to understand anything, because I don't think going through a step by step of deductions actually leads to understanding. But it establishes perfect rigorousness such that you _know_ that you are correct. You can then correlate this with your intuition to _test_ your intuition, and, over time, proofs can make your intuition stronger and more accurate. But the 'proof itself' isn't providing anything in terms of intuitive knowledge; it is correlating whether your intuition is right or wrong that allows you to discard wrong intuitions
@@pyropulseIXXI Yeah, this is pretty similar to how I used to view proofs throughout my undergraduate studies, even a bit into my phd studies. Then things changed a few years back as I got more interested in education more broadly and the meaning of aesthetics in mathematics (also, some of my phd supervisor's views on things have rubbed off on me over the years, but that's just par for the course). Some of this affects the way I view proofs, not just as a logical set of deductions, but as a thing of potential beauty if done correctly. Nowadays I really like to try and discover "elegant" and "crystal-clear" arguments to prove a statement, since I find these both appealing and of high pedagogical value (of course, I don't always succeed to find these kinds of arguments, but that's another story entirely). With regards to "intuition" I think there can be an interesting discussion about the semantics of the word, but I do think I understand where you're coming from. Interestingly, if we consider the addition/subraction formulas for sine and cosine, I think the most "intuitively clear" one is actually the difference formula for cosine (which is coincidentally the first thing I proved in this video). This is because of the fact that this particular formula can be seen as the scalar product of two unit vectors, making it completely trivial to calculate once we have a firm understanding of the underlying concept. I decided not to even bring this point up in the video, but I'm almost convinced that the argument I showed in the "ugly" proof was actually inspired by linear algebra to begin with, only to be phrased in a way as to completely remove any trace of the original source of intuition behind the argument...
I suppose mathematics can inspire a certain religious veneration, but "awesome", in addition to being a word virtually threadbare with abuse, is a superlative which should admit of no qualification. It might be enough to use the more sober "interesting".
While I am reminded of the religious connotations to the word "awesome" now that you mention it, that was certainly not the interpretation of the title that I was going for. In fact, this meaning is something that most non-natives learn quite late in their English education (if at all) and thus it's far from the first thing I think about when I hear/see the word. Mostly I think of "awesome" in the colloquial sense, where it is not considered a superlative but rather just a strong word for "good/great".
This video lesson is ruined by the background 'music' - distracting, irrelevant, making it difficult to hear the spoken words, and difficult to concentrate on the content. Why do programme makers assume their audience have a concentration span of only 3.5 seconds unless woken up by 'music' ?
I'm sorry that your viewing experience was ruined by the background music. The reason I added the music was that I felt it made the video better (a decision I made after syncing up the narration with the animations, and comparing the video with vs. without bg music), and I did try to mix it so that it wouldn't interfere with the narration. While it is unfortunate that some may find it distracting I still stand by my decision to add music to the final product. And just to be perfectly clear: I didn't add the music as a cheap attempt at "fanservice", or because I believed people watching would have deficient attention spans. I added the music because I felt it was the right thing to do in order to raise the quality even further on a passion project I had been working on for a long while.
Good question. I wanted to make this video as accessible as possible since this was a SoME contribution and I know that there are many people who are interested in the event that aren't very familiar with vectors to begin with. So the vector explanation part isn't really aimed at people who already know vectors in and out, which is why I thought it was reasonable to mark out sections clearly so that people could skip ahead more easily. In either case, I don't think it hurts to include an explanation of these things in the video even if most people who will see the video already know what I'm talking about.
Very nice video. I am a hobbyist and do a lot of earth sun geometry etc. videos using Geogebra so this is quite applicable. I also wondered about the introduction to vectors being included but I acknowledge the point you made in a previous reply. Anyway very well done thank you
@distro logic ... It doesn't hurt much if u get a different presentation of something u already know about.... Ig everyone might have had their share of insights (at least i got it) after watching this and remember presentation is an art in itself... No matter how much u know if u can't present them properly u amount to nothing... So gotta appreciate the dude at least for his beautiful exposition and illustrations.
6:25 "After all, this proof uses some rather ad-hoc methods, seemingly pulled out of thin air and just happens to work out at the end" Mmmm, you clearly have not had much exposure to maths. Much of maths is 'ad hoc' and 'just happens to work out'.
I don't know if this is a troll comment or not, but I'm going to treat it as if your message is sincere. I don't appreciate your dismissive tone, pulling a quote out of context while completely ignoring the surrounding argument I was trying to make. I never said anything about ad hoc methods never being used in mathematics in the video, but rather my argument is that if a proof uses these methods then there is a very good reason to look for a better proof, as there is probably some aspect of the problem that is yet to be understood if all the proofs that we have are completely ad hoc in nature. I thought that I stated my position well enough in the video, but as your comment indicates maybe that was not quite the case. And btw, just to be perfectly clear about my exposure to mathematics: I'm a phd student in pure maths, currently in my last year before defending my doctoral thesis.
Ok, this is my favourite #SoME2 as it is in the spirit of the original 3blue1brown videos - taking an obscure memorised algebraic fact and turning it into a statement of visual obviousness. Kudos.
Thanks, I'm glad you liked it :)
We actually never learned the formulas for cos(x + y) or sin(x + y) in normal maths class. I'm now in my last year of hs, and I chose to participate in an optional maths course, where we learned about these formulas using complex numbers (10x more elegant imhu).
If you didn't already know, the proof goes as follows:
(Note that I don't have the greek letter theta, so I'll use x)
e^(i(x + y))
= e^(ix + iy)
= e^(ix) * e^(iy)
= (cos(x) + isin(x))*(cos(y) + isin(y))
= cos(x)cos(y) + cos(x)*i*sin(y) + isin(x)cos(y) + isin(x) * isin(y)
= ( cos(x)cos(y) - sin(x)sin(y) )
+ i( cos(x)sin(y) + sin(x)cos(y) )
(Split into real and imaginary parts)
As such, we have the formula for cos(x + y) in the first term, and the formula for sin(x + y) in the second by identification, since
e^(i(x + y))
= cos(x + y) + isin(x + y)
This proof is equivalent to the one I gave in the video, with the only difference that it presupposes some nontrivial machinery for it to actually work out.
The main reason why I avoid this approach is not because it's invalid or difficult to explain once you know the properties of the complex exponential function. Rather, the difficulty lies in explaining why the complex exponential function works the way it does.
For instance, can you give me a valid explanation for why e^(ix) should equal cos(x)+isin(x)? And furthermore, how do you prove that this definition gives us an exponential function that satisfies the equation e^(ix+iy)=e^(ix)*e^(iy) for all x and y (without already knowing the addition formulas for sine and cosine, that is)?
These questions are not necessarily easy to answer, and I would expect most people to struggle with them quite a bit, even with some explanation given.
This is, in part, why I chose to prove the statement the way I did. One of the "downsides" of my approach is that I need to spend some time to lay the groundwork explaining what 2d vectors are and how rotations of them work, but once I have that the proof is basically just one line. The upside of my approach, however, is that all the preprequisite linear algebra that I need to explain it is elementary, and it's simple to explain intuitively with the use of graphical aids.
Personally, I like using complex numbers to derive the angle sum formulae because they follow directly from the dead simple power sum formula: e^(a+b) = (e^a)(e^b). Using Euler's formula e^ix = cos(x) + isin(x), you can then do e^i(a+b) = (e^ia)(e^ib) = (cos(a) + isin(a))(cos(b) + isin(b)) = cos(a)cos(b) + icos(a)sin(b) + isin(a)cos(b) - sin(a)sin(b). Separate the real and imaginary parts and you get sin(a+b) = cos(a)sin(b) + sin(a)cos(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b). This also follows the geometric interpretation used in the video since multiplying by a complex number is scaling by the magnitude (which is 1 in this case) and rotating by the angle, so you're really just rotating another rotation.
Regardless of how you do it, the truth will pretty much always be that sine and cosine belong together and it's so much easier to handle them together rather than separately. sine and cosine are just two halves of a whole, the whole being one of the simplest operations in geometry: rotation.
Yeah, multiplication of complex numbers is a great way to derive these formulas. The only slight problem I have with this method is that you kind of already have to understand why e^(a+b)=(e^a)*(e^b) still holds when a and b are complex numbers. Many times that identity is proved by alredy knowing that the addition formulas for sine and cosine hold, which leaves us with a bit of a conundrum. But in case that I already know that the formulas are true, then I think that using this method is probably the easiest way to derive the formulas if one should forget them.
And I totally agree when you say that sine and cosine belong together. They are really just two sides of the same coin.
@@TheArmchairIntellectual I thought that the power law was a part of the definition when extending to non-natural number exponents, complex or otherwise, with e^x specifically having the extra requirement of d/dx e^x = e^x.
With the latter, it's possible to get an idea for the complex exponential even without using euler's formula. d/dx e^ix = ie^ix via chain rule. i itself is a rotation by 90°, so the derivative of e^ix is always perpendicular to the function itself. The main area I've seen that property is in rotational dynamics, where the speed of an object remains constant, but the velocity gets accelerated in a perpendicular direction toward some center.
@@angeldude101 As far as I'm aware the exponential function is usually defined using its power series representation so that it can be extended to general Banach algebras, and then properties are derived from the series expansion. But yeah, there is no real issue having the power rule as a part of the definition of the exponential function, and then basically just use a differential equation to seal the deal; the necessary boundary condition e^0=1 seems to follow automatically from this meaning that it becomes redundant in a sense... (one can of course use the differential equation + a boundary condition as well, but then the power law has to be proven). I've actually thought a bit about how to use the power rule without using differential calculus to define the complex exponential, and I have some ideas of maybe making a video about it in the future if I can think of a good way to present it in the way I want to.
In either case, no matter which way we use to define the complex exponential there is some explaining that needs to be done before it's obvious that e^{ix}=cos x + i sin x (using varying degrees of differential calculus), and I kind of felt like no matter which explanation I used it wouldn't be possible to showcase the main thing I wanted to in this video; after all, the idea I wanted to showcase was the use of a dictionary to solve a problem in an unexpected way, which is an approach to problem-solving that I just find super appealing.
This works too because the complex plane is essentially the same as (there exists a bijective mapping to) R^2
its fun until we realise there is Circular reasoning.
Euler's formula requires taylor series
taylor series requires differentiating sin, cos, exp
diff sin requires evaluating limit (sin(x+h)-sin x)/h , h->0
that limit requires sum angle formula which requires euler's formula :(
cool vid. i thought you'd encode rotations as matrices and use that to rotate the vectors but i was pleasantly surprised to see that you thought about rotations more abstractly and intuited what useful algebraic properties it would have and how the angle sum formulas would follow. i would've been happy to see this when i was first learning about the compound angle formulas because it feels nicely motivated and like an obvious consequence of some properties about rotations unlike the standard way it's taught.
I completely agree. The actual reason I came up with this proof about two years ago was precisely because of the pedagogical potential I saw in it. I was taking a teaching course at the time, and it really got me thinking about a few things, among them this video idea.
@@TheArmchairIntellectual It feels like you did the matrices without doing the matrices! Personally, although I think not bringing matrices into it was by far and away the right call, it might’ve been worth flashing up a brief “for the interested” on screen just showing the proof in matrix multiplication form. If it goes over your head, fine; if either you know matrices or can figure it out, then excellent. Still a great video, and made my jaw drop.
@@krozjr5009 Yeah, after all linear maps and matrices are the same thing, so we can definitely say that I talked about matrices without mentioning them. I just felt like it would take too much effort to explain them, without really making the core points of the video any clearer in the end, so I decided that things would work out just fine if I didn't even mention them :)
Sick pfp
the rotational aspect which dissolves some of the most "difficult" questions on matrices, and no one seems to teach it except online (at hs level)
Hmm I didn't learn that 'ugly' proof, I learned a different one where you draw right triangles inside a rectangle. Granted it only works for acute angles but it's at least more motivated and less symbol-shuffley than that one. But even so, the vector proof is so much nicer! Great video!
I've never seen that proof but yeah, at this moment it seems to me like we will lose a lot if we don't use rotations to justify the addition formulas, no matter how we try to avoid them. And this is probably not very strange considering how closely tied sine and cosine are to rotations.
That wouldn't be as ugly
Yeah I definitely also learned some geometric mumbo jumbo. Spent a lot of days trying to make enough intuitive sense of the steps to be able to reconstruct it on my foggy shower wall.
This is how this was taught in Poland 2007-2010. I remember that class: loads of chalk, plowing through identities and final arrival at concise and satysfying formula. Fun times. It absolutely can be entertaining.
This felt like an expansion of the Essence of Linear Algebra series, it was so good! As an educational video creator myself, I understand how much effort must have been put into this. Liked and subscribed, always enjoy supporting fellow small creators :)
Thanks for the appreciation, I'm glad that you liked it :)
This was the best IMO. Awesome video, as I nearly forgot all of trigonometry, helped alot. Keep going!!
Where was the IMO problem
@@好吧-h6k IMO: in my opinion
he said that this video was the best, in his opinion
Call me masochist, but I enjoyed the torture section...
This is actually helpful
That's the first time I saw sin and cos as Cartesian coordinates of a unit circle!
Back in trig and pre-calc I asked my teacher what was the formula inside the sin function , both teachers just said to look it up on the table or use a calculator, I never got any understanding from that☹️
That's unfortunate really. Viewing sine and cosine as the coordinates of the unit circle is by far the most common way to geometrically describe them for arbitrary angles, and it really helps in understanding why they are so useful when considering rotations.
@@TheArmchairIntellectualThis is the first time I've seen in and I saw math courses from multiple universities not describing it that way... Probably because they expect teacher to teach basics, but they did fuck all.
After some reading on sinus and cosinus I just learned that I only knew the definition for triangles... I didn't know the definition for circles even existed. WTF now I can go back 2y and finally understand some of the lectures I just could not comprehend....
The art of math is to see the bigger picture of any mathematical structure one is trying to construct. The way you started with the definition of a general vector (to have operations that can create new elements using pre existing elements, not mentioning the exact wording but essence remains the same) and then developed necessary motivation needed to create a geometric vector, correlate algebraic vectors to geometric counterpart so that the correlation is one to one (aka unique) was breathtaking. My mind went boom and said eureka (though instead of finding elegant solution of some problem it was due to coming across this video). Excellently explained , ur voice was a cherry on the topping. Hope u go a long way. And don't stop making videos. Want to relish more of these mathematical delicacies.
I think this is my favorite #SoME2 video so far
Ok this is absolutely BEAUTIFUL
Thanks for explaining everything thoroughly, even if a lot of those things are obvious to math enthusiasts. A lot of people just skip over the explanation and take stuff for granted.. That's missing the whole point of these videos!
Yeah, it's all about the intended audience. And here I didn't want people unfamiliar with trig and linear algebra to feel completely left out, so I decided to include stuff that some people might find a bit superfluous.
YOU ARE A GOD... NO QUESTION ABOUT IT
Hmmm, I'll take that as a very enthustiastic compliment :)
I loved it ! Beautiful video and proof ^^
Gonna use this for a proof
My favorite way of proving those trig identities is using complex exponentials. It's so easy and simple, I don't even memorize the relations, just work with complex exponential whenever I need them
Of course, this is a very simple way to do it once we fully understand how the complex exponential works (and how it is connected to rotations in the complex plane).
My only problem with this kind of proof is that Euler's formula e^{ix}=cos x+i sin x is normally just stated rather than fully explained when first introduced, and the fundamental properties of the complex exponential tend to be proven using the addition formulas for sine and cosine. So unless we already have an independent understanding of the complex exponential and how it connects to rotations, we end up with a sort of circular argument.
I hope your channel grows bigger soon! This was helpful!
Lol and I hope I grow too, my subscriber goal is 10 000 000 in 1-3 months
We'll see how it goes, no matter what happens next I'm just super happy both with what I managed to make and that people seem to actually appreciate it. If nothing else, the entire experience surrounding this video has been a huge confidence boost :)
The idea that we intuitively knew about but never got the time to prove
Thank you! Everything can be seen as a vector!
the proof i learned (well, found and remembered) was to construct line segments with angle theta and (theta+phi) over a unit circle, then construct lines to form the right triangle from the endpoint of the latter to the former, and from that intersection point to the x axis. extend the latter line vertically, and construct another line to form a third right triangle with the first point mentioned.
essentially, we have constructed a right trapezoid with the diagonal line having length 1, sectioned into 3 right triangles by 2 lines intersecting at a right angle somewhere along the far edge. that edge, in fact, has the length sin(theta+phi), but it is split into two sections with lengths we can calculate: the right triangle from theta to (theta+phi) has hypotenuse 1 and feet of lengths cos(phi) and sin(phi). those feet are the hypotenuses of the other 2 right triangles.
those other 2 right triangles must be similar to each other because, if the first one has angles a, 90, and b, we know that a+b=90 by the triangle inner angle sum, and we know that b+90+n=180, where n is the first angle of the second right triangle in question. therefore, n must be a, and the other angle must be b.
we know that a is theta because it is the origin angle of the right triangle defined by the x axis and the line with angle theta from it. therefore the full far edge, sin(theta+phi), is equal to the sum of its two partitions, which, as is clear from the construction, have lengths cos(phi)sin(theta) and sin(phi)cos(theta).
this proof is surprisingly difficult to put to words alone, but it is extremely simple visually, especially if you take it for granted that the outer two of the right triangles are similar. you literally just draw the right trapezoid sectioned into the three right triangles, and the angle sum formula falls out.
Do you perhaps have a link to this proof, where it's accompanied by a visual explanation of the line constructions involved? I think that I'm kind of messing it up somewhere when I try to construct the lines you describe...
Anyway, thanks for providing a different way to prove these formulas, I really like it when there are multiple different ways to prove the same thing :)
@@TheArmchairIntellectual i made a reply, but it seems to have been automatically deleted, likely for containing a link. in place of the link, here's instructions to the same end: open up a desmos calculator tab and append "/wpaptamvod" (without the quotes of course) after /calculator. it should contain the construction, together with some notes and two sliders to adjust the angles (limited to acute positive angles, as that range is where the identity being proven is most clearly shown by the construction--and I didn't want to keep having to think about the bounds of the equations that would be needed so that the lines would display properly for all cases with obtuse angles)
one thing that i just noticed: t and p are expressed in radians on the sliders, but i wrote the angles in the notes in degrees. what i mean should be fairly clear despite this.
@@rarebeeph1783 Thanks a lot!
Not so painfull friend! It would be great to study all demos this way!
Nice video. Your method using vector is really nice. My preferred method is using complex numbers and Euler's formula, which reframes the problem into one of purely algebra.
Yeah, I think that using Euler's formula is maybe the easiest way in practice once you understand why Euler's formula actually works (but I didn't want to explain that in this video :)), since it doesn't require us to build a formula for rotations from scratch. So in a pinch, if you forget what the formulas are then using that method is really convenient. In a way though, that proof is very similar in spirit to the one I gave since the core of both proofs really is exploiting rotations in the plane (whether we view it as complex or not).
My favorite example of the power of vectors is Bottema's Theorem. Look it up!
Excellent video. Just a minor comment I want to make and I hope it doesn't come off as snarky but I would like to add my 2 cents.
I wouldn't really call the trig identity for sum of angles or ANY trig identity for that matter a "proof" but rather a derivation.
They exist and are defined as such because of so.
If someone doesn't believe a trig identity, then you could "prove" it to them by showing them a logical derivation which you have done with perfection.
Good stuff. Make more.
I like the idea that a proof is only really useful if it's comprised of obvious steps. Of course there are proofs that can't be simplified to fit this category - and that's the thing, these obscure proofs feel more like axioms where you either know the result or don't, without really being able to prove it yourself.
I think the best kind of proof is the one you can fall back on in case you forgot the relation in hand. In the best case you wouldn't even have to learn the result. Like here, with the addition and multiplication identities of the trig functions, it's way easier to do the 5 minute derivation, with linear algebra for example, than to recite these seemingly meaningless formulas.
I wouldn't go so far as to say that a proof is only useful if it's comprised entirely of obvious steps, rather that if we have a proof that isn't then there is probably something about the statement that we have yet to understand fully. I mean, most of the time mathematicians will initially find a proof of a theorem that is really complicated and ad hoc, but that proof usually serves as a starting point for actually dealing with the statement with more confidence.
In either case, I think there is a lot more to mathematical proofs which often gets overshadowed by the statements that they're meant to prove.
29:39 Is it just a coincidence that the x-coordinate of the rotation of the vector [a, b] by an angle theta is precisely the derivative with respect to theta of the y-coordinate of the same rotated vector? I'm gonna have to look into this on my own... Oh and great video btw :) I'm taking linear algebra right now so this is really helpful and interesting.
Hmm, I haven't really thought about it but there is probably some way to explain why that is the case. Maybe has something to do with the complex exponential function, but I don't know at the moment.
I guess it might be related to something on the lines of this
If we imagine an object rotating along a unit circle at the rate of 1 revolution per unit time, we note these facts:
1. The velocity is just the rate of change of its position (i.e. its derivative). Hence, *v* = d *r* /dt
2. The velocity vector is perpendicular to its position vectors at any given moment. Hence *v* ⊥ *r*
3. The lengths of position vector and velocity vector must be equal as | *r* | = | *v* | = 1
When object's position was *x* , its velocity was just *y* . Rotating this snapshot by θ just corresponds to Rot( *x* ) and Rot( *y* ). We already noted that *v* = d *r* /dt. So *v* = Rot( *y* ) is just d/dt (Rot( *x* )). So I guess this somehow shows y component being the derivative of x component.
I guess this is a very crude way to see it 😅. But only this much is coming to my mind at the moment.
Note: I wasn't able to find the cap symbol, so through *x* and *y* I mean x hat and y hat (the unit vectors)
Great!
Wouln't the argument you make in minute 3 just valid for differences under 90 degrees? Also in the final argument, how do you extend the statement that the vector (0,1) going to (-sin(),cos()) to greater angles?
Hi there! I'm not quite sure which exact argument you're referring to at the 3 minute mark. If it's the argument that |PQ|=|AB|, then this argument really doesn't depend on the size of the angle difference at all, as it is only concerned with lenght segments that are rotated around the origin. Feel free to comment if I misunderstood which argument you referred to here.
As for your second question, this is something that is easiest done by just checking manually for the four different nontrivial cases (angle between 0 and 90 degrees, between 90 and 180 degrees, between 180 and 270 degrees, and between 270 and 360 degrees); after this everything works on account of periodicity. This can be done in each case by making basically the exact same argument that I gave in the video for each quadrant, and checking that the signs really are correct in each case. One could conceivably try to find a more elegant argument here to deal with general angles without checking different cases, but I think that would be a waste of time tbh.
I think we proved these identifies using the complex exponentials in university.
While this is certainly possible, it's trickier than it seems at first glance if one wants to avoid circular logic. Basically, one has to motivate why e^(ix)=cos(x)+i sin(x) and then independently motivate why e^(ix+iy)=e^(ix)e^(iy) (of course without using the trig identities that we wish to prove).
At the end of the day I only see two general approaches one could take. Firstly, we have the geometric approach where one argues using rotations in a manner similar to what I did in this video.
Secondly we have the differential approach, where one could work using Maclaurin/Taylor series expansions or differential equations. However, this second approach requires some calculus know-how before one could hope to pull it off, and it really seems like overkill to me when trying to prove some trig identities.
324 th subscriber here and I love maths very very very much...
3:42 awesomeeeeeeeee
Hmmm, I have to say that I didn't intend for what is happening at this time to be considered as "awesome", but I guess I'm glad someone enjoys this part as well...
This was fun~
I'm glad you liked it :)
This is a good video but I doubt some would accept a topic so easy
I might be masochist cuz i like the torture session
What would a row vector represent in this context?
In general, row and column vectors are dual to one another, so a row vector in this context would correspond to a 1-form. I think 3blue1brown talked about this in his EssenceOfLinearAlgebra series in connection with the dot product. My opinion is that the dual of a linear space can be a bit difficult to motivate from an intuitive point of view unless one is already familiar with linear algebra to begin with, so this is part of the reason why I chose not to cover this nuance in the video.
What do i have to learn in order to understand all of this? Any guide ? :)
I guess that depends a bit on exactly what you find most troublesome to understand. I tried to make the video accessible to anyone who is familiar with high school mathematics (like some trigonometry, a decent grasp on the algebra of real numbers), but of course that can be a bit difficult for me to do perfectly since it was such a long while since I was in high school.
Anyway, I think it would be easier for me to answer your question if you could point to something specific in the video that you found difficult to follow along with.
Hi,
If Cartesian Coordinate System gives just scalar multiplication to basis vector (i, j).
What are polar coordinates doing... In which one in just scalar distance(r) and the other angle is?..
Which type of tensor is angle in the set of (distance, angle, area volume).
Could you make sense of this....
I'm not quite sure what it is that you are asking to be perfectly honest.
As far as I'm aware, polar coordinates are not commonly used when talking about linear algebra, but if you really wish to connect "polar coordinates" (where we don't have the zero element well-represented) with some sort of "tensor" I suppose your best bet may be to consider the [angle] part of the polar coordinate be a rotation of the plane (i.e., it's an element of the group of linear transformations of the plane called SO(2)), and to consider the [distance] part to denote a scaling operator, which scales vectors in the plane by a factor [distance].
What do you mean by zero element not defined..
@@kartikeyedunite You don't have polar coordinates well-defined for the complex number 0, nothing more complicated than that.
What if we define vector addition like this
V¹(r¹, θ¹) + V²(r¹,θ²)= V³[(r¹+r²),(θ¹+θ²)]
Sorry if this is nonsense..
Thanks a lot for your reply 😊..
@@kartikeyedunite Of course, you could make this definition. But then you would lose out on the regular properties of polar coordinates that one typically associates with them. Basically, with this definition you would end up with polar coordinates working like regular cartesian coordinates from an algebraic perspective, and it would have nothing to do with the standard way we actually add complex numbers.
In other words, it usually doesn't work to try and have your cake and eat it too.
Wait!! Are you saying that math can be easy and simple if you looked hard enough?? That's cool
Sure am! In fact, this is the real difference between a novice and an expert in almost any topic, not just mathematics; the expert has just looked hard enough at enough things to know how to make things easier for themselves. While a bit of a simplification, I think it's good to keep this in mind, since so many people tend to overestimate the importance of talent and ignore the amount of hard work that goes into mastering anything.
nice proof! I can't help but ask; since you said vectors are "things that we can add up and scale to get the same things" don't rotations also check out? You gave the addition and scaling rules in the video. Im a 4th year CS student, so you can assume some basic knowledge :)
We usually think of rotations in the plane as forming a structure called a group, with the group operation being composition. Since they are linear maps, they can be identified with a set of matrices, often denoted by SO(2,R); this is a subgroup of GL(2,R) often called the special orthogonal group. Then rotations can be viewed as matrices and the group operation becomes matrix multiplication. However, I haven't really heard this group described as a vector space, since it's not clear how one is supposed to scale a rotation and get back a rotation afterwards; since the group is abelian we may view the group operation as "addition" if we want to, but the issue with scaling by real constant is something I don't know how one would resolve.
What's your vector, Victor?
do you have 3D version of this?
Not on this channel as of yet (and I'm quite frankly not entirely sure what a 3D version of "this" would be).
I have some projects I'd like to cover on this channel in the future (although it will have to wait until after I finish my PhD), we'll see if I find something interesting to talk about in 3D linear algebra. I'm not sure what topic that would be at the moment, but who knows what the future holds?
Sorry, javascript burn my brain. I mean vector in 3D or N dimensions. How it rotate, and translate in 3D space.
you did a great job! thank you thank you
@@lkda01 I see. If we discuss rotations in higher dimensions this is usually generalized to groups of linear transformations called the special orthogonal (or unitary in the case of complex numbers) groups, which is an interesting topic to discuss.
When it comes to translations I guess it may be best to discuss the matter in terms of affine transformations, which are transformations that are "almost" linear. While I'm no expert on affine transformations it may be interesting to go over sometime in the future.
how u animate this?
I used the python library called ManimCE (the community edition of Manim that was created by Grant Sanderson). It took some time to animate, but I think the end result was worth the effort.
i am not math oriented.....as a college student i found the trig functions to be unpleasant and easy to forget or confuse.....it was like learning a new language, but i found studying russian and french were more congenial and less mechanical than mastering logarithms..........................in a way its sad that so much of the important industry depends on higher math because probably 97% of people find math to be intimidating and unpleasant, and no matter how many problems you solve and master there is always another problem ahead that is worse than the problem you just laboriously wrestled with for 2 hours.
how do you square a distance? how do you square an orange? how do you square an angle? how do you square a '57 chevy? how do you square a trombone? how do you square a dog?
i understand how you square a number ...... a distance from miami to atlanta is not similar to the number 12 ...... you can fly an airplane to atlanta, you cannot fly an airplane to the number 12 ....... 12 is an idea in the mind ..... distance is a physical separation of two locations
This proof still doesn't make it intuitive, because that isn't what proofs do. Like, no crap that rotating leads to that formula, but the specific form of that formula itself is not made intuitive depending on the proof you do, and you still have to work the proof to see what form the angle summation formula(s) take(s)
If that form was intuitive, I would be able to write the form just by looking cos( a + b ) and knowing intuitively that it equals cos(a)cos(b) - sin(a)sin(b).The moment you do any work, it isn't intuition anymore (however, knowing exactly how to proceed to arrive at the solution or proof form is also a form of intuition, but that means the two proofs you showed are not better than each other in any respect; in fact, intuition allows one to proceed with either proof in the correct manner and on the first try)
I'm not sure how to respond to this comment in the best way. While I agree that it's never going to become intuitively obvious that the formula for cosine is exactly what it is, if we understand rotations as linear transformations it is pretty intuitively clear that there should be a formula for the sine and cosine of a+b using the sines and cosines of a and b.
So while it's not going to be completely intuitively obvious that cos(a+b)=cos a cos b- sin a sin b, it can make perfect intuitive sense that the formula must look very similar to this, and then finding its exact form is more of a formality than a challenge.
And I'm not entirely sure what you mean that proofs "do", exactly. There are many proofs out there that are both hard to follow and will not make it clear at all why something works on a conceptual level, while there are other proofs that basically explain why the statement they are supposed to prove make perfect sense. While the basic requirement for a proof is that it is a logical derivation of a certain fact, this is not the only thing a proof can be good for, and there is a lot of difference in a "good" and a "bad" proof, even when both proofs are perfectly logically valid.
At the end of the day, while I do have some disagreements with you I do really appreciate that you took the time to explain your perspective (it really helps me to see why someone might not view my video in the same high regard that I do; while I'm trying to be objective about it it's always going to be hard not to be somewhat biased...).
@@TheArmchairIntellectual I mostly agree with everything you just said. Except when I think of 'intuitive' I think I can immediately understand it and know the form with my intuition alone. If I have to do any work at all, then that isn't intuition anymore.
There can be intuition in knowing how to proceed such that the work that you do leads to the right answer on the first try, or in the right direction without much fuss, such as knowing to use rotations to lead to an answer proof of angle addition formula.
But I still cannot say that the formula itself is intuitive; just that rotations makes doing the proof intuitive.
Also, you say I may not view your video in the same high regard, but I greatly enjoyed this video and think that it is of top quality
I dual majored in physics and mathematics, so I don't know if my conceptions are different, but a proof is a logical set of deductions that establishes a truth
I am of the view that all of mathematics is just one big tautology. Or at least the stuff that is consistent is one big tautology; whether it is complete or not is irrelevant
I've never really used proofs to understand anything, because I don't think going through a step by step of deductions actually leads to understanding. But it establishes perfect rigorousness such that you _know_ that you are correct.
You can then correlate this with your intuition to _test_ your intuition, and, over time, proofs can make your intuition stronger and more accurate. But the 'proof itself' isn't providing anything in terms of intuitive knowledge; it is correlating whether your intuition is right or wrong that allows you to discard wrong intuitions
@@pyropulseIXXI Yeah, this is pretty similar to how I used to view proofs throughout my undergraduate studies, even a bit into my phd studies. Then things changed a few years back as I got more interested in education more broadly and the meaning of aesthetics in mathematics (also, some of my phd supervisor's views on things have rubbed off on me over the years, but that's just par for the course). Some of this affects the way I view proofs, not just as a logical set of deductions, but as a thing of potential beauty if done correctly. Nowadays I really like to try and discover "elegant" and "crystal-clear" arguments to prove a statement, since I find these both appealing and of high pedagogical value (of course, I don't always succeed to find these kinds of arguments, but that's another story entirely).
With regards to "intuition" I think there can be an interesting discussion about the semantics of the word, but I do think I understand where you're coming from. Interestingly, if we consider the addition/subraction formulas for sine and cosine, I think the most "intuitively clear" one is actually the difference formula for cosine (which is coincidentally the first thing I proved in this video). This is because of the fact that this particular formula can be seen as the scalar product of two unit vectors, making it completely trivial to calculate once we have a firm understanding of the underlying concept. I decided not to even bring this point up in the video, but I'm almost convinced that the argument I showed in the "ugly" proof was actually inspired by linear algebra to begin with, only to be phrased in a way as to completely remove any trace of the original source of intuition behind the argument...
I suppose mathematics can inspire a certain religious veneration, but "awesome", in addition to being a word virtually threadbare with abuse, is a superlative which should admit of no qualification. It might be enough to use the more sober "interesting".
While I am reminded of the religious connotations to the word "awesome" now that you mention it, that was certainly not the interpretation of the title that I was going for. In fact, this meaning is something that most non-natives learn quite late in their English education (if at all) and thus it's far from the first thing I think about when I hear/see the word. Mostly I think of "awesome" in the colloquial sense, where it is not considered a superlative but rather just a strong word for "good/great".
0:19
If you like trigonometry, you haven't done enough trigonometry to find out that you actually don't.
This video lesson is ruined by the background 'music' - distracting, irrelevant, making it difficult to hear the spoken words, and difficult to concentrate on the content. Why do programme makers assume their audience have a concentration span of only 3.5 seconds unless woken up by 'music' ?
I'm sorry that your viewing experience was ruined by the background music. The reason I added the music was that I felt it made the video better (a decision I made after syncing up the narration with the animations, and comparing the video with vs. without bg music), and I did try to mix it so that it wouldn't interfere with the narration. While it is unfortunate that some may find it distracting I still stand by my decision to add music to the final product.
And just to be perfectly clear: I didn't add the music as a cheap attempt at "fanservice", or because I believed people watching would have deficient attention spans. I added the music because I felt it was the right thing to do in order to raise the quality even further on a passion project I had been working on for a long while.
Who getting this in their recommended or even searching for this doesn't know what vectors are..?
Good question. I wanted to make this video as accessible as possible since this was a SoME contribution and I know that there are many people who are interested in the event that aren't very familiar with vectors to begin with. So the vector explanation part isn't really aimed at people who already know vectors in and out, which is why I thought it was reasonable to mark out sections clearly so that people could skip ahead more easily. In either case, I don't think it hurts to include an explanation of these things in the video even if most people who will see the video already know what I'm talking about.
Very nice video. I am a hobbyist and do a lot of earth sun geometry etc. videos using Geogebra so this is quite applicable. I also wondered about the introduction to vectors being included but I acknowledge the point you made in a previous reply. Anyway very well done thank you
@distro logic ... It doesn't hurt much if u get a different presentation of something u already know about.... Ig everyone might have had their share of insights (at least i got it) after watching this and remember presentation is an art in itself... No matter how much u know if u can't present them properly u amount to nothing... So gotta appreciate the dude at least for his beautiful exposition and illustrations.
6:25
"After all, this proof uses some rather ad-hoc methods, seemingly pulled out of thin air and just happens to work out at the end"
Mmmm, you clearly have not had much exposure to maths. Much of maths is 'ad hoc' and 'just happens to work out'.
I don't know if this is a troll comment or not, but I'm going to treat it as if your message is sincere.
I don't appreciate your dismissive tone, pulling a quote out of context while completely ignoring the surrounding argument I was trying to make.
I never said anything about ad hoc methods never being used in mathematics in the video, but rather my argument is that if a proof uses these methods then there is a very good reason to look for a better proof, as there is probably some aspect of the problem that is yet to be understood if all the proofs that we have are completely ad hoc in nature.
I thought that I stated my position well enough in the video, but as your comment indicates maybe that was not quite the case.
And btw, just to be perfectly clear about my exposure to mathematics: I'm a phd student in pure maths, currently in my last year before defending my doctoral thesis.