Rotation matrix derivation (step-by-step prove)

👉 In this video we derive the Rotation Matrix that represents a coordinate transformation by rotation over an angle.
🚀 Related Topics:
Matrix Multiplication: • Vector Algebra - Exerc...
👩‍🔬 If you have any questions, remarks or suggestions, leave a comment :)
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#RotationMatrix #Stepbystep #Derivation

[@PenandPaperScience]

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[@PRIYANSH_SUTHAR]

I think if you only want to rotate the coordinate system by alpha, you can multiply Matrix of unit vectors of original system by rotation matrix. And then as someone said before, multiply the new rotated vector by the inverse of the rotation matrix to get the vector back.

[@PenandPaperScience]

Yes, you can also do that. I even think it boils down to the same thing. (: What you're doing in your scenario is basically transforming a matrix, instead of two vectors.@@PRIYANSH_SUTHAR

[@liquidportal9082]

By far the best derivation for rotational matrices on KZbin. Excellent work.

[@PenandPaperScience]

Thank you for the kind words! I'm glad I could help :))

[@anonymanonym5901]

Wow, thank you so much, this video is awesome. I watched so many videos about rotation matrices and I was so confused about how you compute the x' and y' (I didn't know if you multiply x' by everything in the rotation matrix and then what abt x and y, etc.). But now I understand it's split into rows. Thank you so much!

[@PenandPaperScience]

You are very welcome, I'm super happy the video was helpful :)

[@EnduroDEV]

Thank you, I was looking for exactly this kind of explanation.

[@PenandPaperScience]

Awesome to hear! It's my pleasure :)

[@godslain9941]

Wow! Ive never heard anyone actually using the correct pronuciation for plus and beta like that! As an italian and a greek/latin student I really appreciate that!

[@PenandPaperScience]

:D That's a nice comment to get, thanks! (Belgian btw)

[@yobniares]

as a russian i appreciate that too :^)

[@chinnuchinnu966]

Best mathematical explanation about rotation matrix, but i miss intuion bro 😢

[@PenandPaperScience]

You found the example video yourself already! (: kzbin.info/www/bejne/n6G3lp6mp56erpo

[@chinnuchinnu966]

@@PenandPaperScience thanks bud 😀

[@breakdancerQ]

Didn't understand about cos/sin identities yet, but when I got that and came back to this I finally understand the derivation of this whole thing!

[@PenandPaperScience]

Awesome! That's what learning mathematics is about (:

[@duckymomo7935]

they didn't teach this/mentioned this at all in trig classes that angle sums is the rotation matrix

[@MOXTHEGAMER]

This is a wonderful video. I tried using my textbook to learn this, but this video made it so easy to understand. Absolutely love this video.

[@PenandPaperScience]

Wow, thanks for the kind words! I motivates me to make more videos :))

[@ilredeldeserto]

so if I understand, here we want rotate a vector counterclockwise keeping the reference system fixed and so I have to use matrix in the video. But if I want keeping the vector fixed and rotate the reference system I have to use the inverse matrix of the matrix showed at the end because it's like to take the vector rotated in the new system and rotate it back clockwise. Is for this reason that there is confusion about where to put the - sign in the sen() of the matrix, it depends on what I want to rotate: the reference system or the vector. Am I right?

[@PenandPaperScience]

You are completely right and put it nicely into words! Nice :)

[@labloom5254]

Thank you, I am studying analytic geometry and this helped a lot

[@PenandPaperScience]

You are very welcome! (: Good luck!

[@ProofDetectives]

Thank you.

[@PenandPaperScience]

You're very welcome! (:

[@ian.ambrose]

Thank you, mister.

[@PenandPaperScience]

You are very welcome! :)

[@harbhajanjutt]

impressive sir

[@PenandPaperScience]

Thank you :)

[@mirok4229]

Finally… many thanks. You saved me!

[@PenandPaperScience]

You're very welcome! Thanks for taking the time to let me know :D

[@realdragon]

Great video

[@PenandPaperScience]

Thank you! (:

[@catmatism]

Love this style. Subscribed!

[@PenandPaperScience]

Thank you! :))

[@user02834]

why tf dont nobody just for once do this with numbers stop fkn yappin and start explaining instead of just repeatedly saying words nobody understands this is so frustrating all channels kinda the same

[@PenandPaperScience]

I can do an example with number if you like. Do you prefer a real-life example, or just one where I fill in a number for the angle and compute the end result?

[@user02834]

@@PenandPaperScience would be really nice if you rotated a square or some 2d shape for better comprehension

[@PenandPaperScience]

@@user02834 Here you go :) kzbin.info/www/bejne/n6G3lp6mp56erpo

[@Jenny-kq9jv]

Hi, I love the video, but I have one question: when I change the x and y values of a function to xcos(b)-ysin(b) and xsin(b)+ycos(b) on Desmos graphing calculator , where b is the degree we want to rotate the function by, the function actually rotates in a clockwise direction instead of anticlockwise like shown in the diagram in the video, and I’m very confused about why?

[@PenandPaperScience]

You raise an excellent point. The difference has to do with your point of view: In this video I show how to derive the rotation matrix if you want to rotate a vector within a fixed coordinate system (x-y). But: if you use this same matrix and apply it to your x and y axes, you are basically rotating your coordinate axis, and not the vector. Now imagine the following: rotating your coordinate axes clockwise over an angle b, then what that equivalently does is rotating any vector (or function) inside that coordinate system *counter clockwise* over the same angle. Does that make sense? :)

[@mz3437]

Thanks for your video - you made it extremely easy to understand :)

[@PenandPaperScience]

Awesome! I'm glad it had the effect I hoped for! :) Thanks for the comment.

[@oussama9040]

Great explanation thank you very much for this video

[@PenandPaperScience]

Thank you for taking to time to comment! I'm super glad the video was useful to you! :))

[@MohammadIsrar-ck2qi]

In Virginia university slides....the trig identity of y' = r sin() sin() + r cos() cos() And here is y' = r cos() sin() + r sin() cos()

[@MohammadIsrar-ck2qi]

Please Guide me

[@PenandPaperScience]

Are you sure that the y' in the slides is the same entity on the graph as the y' in this video? Because what you say for y' is what I have for x', perhaps the axes are different?

[@leandrogcosta]

Very good!

[@PenandPaperScience]

Thanks! :D

[@lmaohaha69]

thank you so much🙏🙏

[@PenandPaperScience]

You are very welcome! (:

[@Kiran77808]

Excellent👍

[@PenandPaperScience]

Thank you :D

[@drallisimo34]

cool tut!!!

[@PenandPaperScience]

Thanks Dan :D

[@ahmetkarakartal9563]

thank you so much

[@PenandPaperScience]

You are very welcome! (: Good luck with your math endeavours!

[@barichm0]

Why use trigonometric identities? The values in the matrix just represent the rotated axis. That is all you need. The trigonometric values are of the angles with respect to the axis, not the point. That is why you don't need to calculate the distance. The explanations I've seen of this are unnecessarily complex.

[@PenandPaperScience]

I don't quite get what you mean. Could you point me to an explanation that is not overly complex like you mention?

[@barichm0]

@@PenandPaperScience What I mean is, you don't need to know the trigonometric identities. If you understand how matrix transformations work, the values are just the trigonometric values of the rotated axis. What they are depends on how the axis are oriented. When I was 13 years old, before I learned the equations, I wrote a computer game that did rotation by calculating the distance and using the arc tangent to get the original angle. That was inefficient, and it had bugs. When I learned the equations, I was confused, because I thought that the trigonometric values were supposed to be multiplied by the radius. The programming book where I read about them didn't even explain how they were derived; it said, "Rather than explain the geometry that derives these equations, we'll look at them from a user standpoint." Many years later, I saw an explanation in another programming book of how they are derived. It explained that they work by rotating the axis themselves rather than the point, but it used polar coordinate rotation and trigonometric identities. I have felt that there should be a simpler explanation. I have since figured it out: The values just represent axis vectors that are rotated and added together.

[@AlFredo-sx2yy]

@@barichm0 in short your comment says "the rotation is obtained by making a rotation". You never really explain where you obtain the formulas from, just that you read them on programming books. Thats cool and all that you can apply highschool level math, everyone go ahead and clap, but like, the purpose of this video is to understand where the rotation matrix comes from so ... yeah?

[@PenandPaperScience]

👉🚀Concrete Example Exercise: kzbin.info/www/bejne/e4vYl5x_rNykfKM

[@marcbennet4346]

concise, complete, accurate: elegant

[@PenandPaperScience]

Thank you so much for the kind words! (:

[@mr.chindo8570]

how to write in terms of alpha instead of beta

[@PenandPaperScience]

Beta is the angle central to the problem: it is the angle over which we perform the transformation (rotation). Therefore, you cannot write the rotation over beta as something with alpha, beta must always be present.

[@rajesh321r]

It would be nice if you can make 3 dimensional matrix as well.

[@PenandPaperScience]

That would mean having two angles, and thus 3 dimensions. This becomes difficult to draw, but maybe I'll try Manim (:

[@xxxhristopher]

This was wonderful 🤧

[@PenandPaperScience]

Thank you! :D Much appreciated :)

[@mr.chindo8570]

in other videos the -sin theta is in 2nd row first column unlike your video where it is in 1st row second column.. any idea why?

[@PenandPaperScience]

Rotation is always relative with respect to the axis, whether you rotate the vector over an angle beta, or you rotate the axis over an angle -beta is the same. Since the sine is an odd function, if you change beta with -beta, the sign in front of the sine changes.

[@rudy7175]

There are two cases to consider. This transformation rotates a vector but keeps the coordinate system unchanged. The transformation you are referring to rotates the coordinate system and doesn’t change the vector.

[@oooooo5262]

my teacher did such a bad explanation in this in linear algebra, thank you sir

[@PenandPaperScience]

You are very welcome! :))

[@ahnaftahmid8219]

thank u sir, was really helpful 😊

[@PenandPaperScience]

My pleasure, I'm glad it helped you! Also, thank you for taking the time to comment :D

[@sonic-ee1dd]

thanks man, i study physics and they always used the rotation matrixes but they never explained them. Thanks a lot for the explanation now i understand how i can know if its an sin or cos.

[@PenandPaperScience]

You're very welcome! I agree that they should do better in most universities! (:

[@ranjankumarsahu998]

Thanks brother

[@PenandPaperScience]

You're very welcome! 👊

[@virozz1024]

Can you do the same for 3x3 ... Please

[@PenandPaperScience]

I will put it on my list! Keep an eye out ;)

[@muthuraj3010]

Top Notch

[@PenandPaperScience]

Thank you! :))

[@BCSFM-ku2fr]

loved ittttt

[@PenandPaperScience]

Awesome! (:

[@aimannafis6555]

if the question says it rotate 30 degrees clockwise does it mean i need to substract it from its origin degree?

[@PenandPaperScience]

Hello! I'm not sure what you mean with origin degree. 30 degrees clockwise would mean rotating over an angle of +30 degrees, and you can fill that in in the rotation matrix.

[@MuhammadSadiq-gz8go]

I am very lucky to found this video what a good explanation

[@PenandPaperScience]

Again, thanks for commenting! I am very happy I could help :)

[@ameliedzinka]

Finally, a normal explanation. Because everybody starts with a vector rotated from 0 angle.

[@PenandPaperScience]

I'm super glad I could help :))

[@ameliedzinka]

Thank you!

[@shamelsinha1626]

super helpful

[@PenandPaperScience]

Thank you for taking the time to comment! I'm glad I could help :)

[@NITianBlood]

Is it applicable to anti-clockwise rotation as well?

[@PenandPaperScience]

Yes, in that can you just apply the exact same dataframe, but change the angle to minus the angle: alpha -> -alpha.

[@NateRiver-h4p]

Very instructive, thanks

[@PenandPaperScience]

You're very welcome :)

[@AryanAgarwal-u7q]

really good video, thank you so much!

[@PenandPaperScience]

Thank you for taking to the time to say that. Really means a lot to me (:

[@shravanchandravanshi9192]

Woow very very very helpful vedio

[@PenandPaperScience]

Thank you very very very much! :D Share the love for science

[@jasonzhang7725]

THANK YOU VERY MUCH!!!!

[@PenandPaperScience]

You are very much welcome! :))

[@povdata]

Best explanation in the Internet!

[@PenandPaperScience]

Thank you

[@cw9249]

thank you. however i was hoping for a more visual proof with geometry of why x' is given by x * cos(beta) - y * sin(beta), or why y' is given by x* sin(beta) + y* cos(beta). i can't find this anywhere. is it possible for you to do this?

[@PenandPaperScience]

You're very welcome, and merry Christmas! I will try and think of a visual proof as I don't know one on the top of my head. If I find one, I will let you know through this comment :)

[@vaibhavgadugale]

❤👍

[@PenandPaperScience]

🙏👌

[@poorianaji3706]

thanks😄

[@PenandPaperScience]

You're very welcome!! :D

[@rumplewang2814]

学会了，太牛逼了，非常感谢❤

[@PenandPaperScience]

I don't read Mandarin, but I can recognise the

[@2fifty533]

makes a lot more sense to me now, thanks

[@PenandPaperScience]

You're very welcome! That's why I make these videos (:

[@2fifty533]

@@PenandPaperScience yeah, from my experience, school just teaches us various formulas and concepts at face value without really going into how they work, which is a shame because there is usually a lot of cleverness and ingenuity behind it but all we see is a magic equation where you plug your numbers in and get stuff out no questions asked so it feels pretty nice to actually have an understanding of the thing that you are working with

[@Rey-zb8el]

What if my starting point is not (1,0) for x ? Is the starting point fixed to be 1,0 ?

[@PenandPaperScience]

I'm not sure what you mean with the starting point. Can you be more specific? (:

[@Rey-zb8el]

I made an error in my calculation and I understand it now. keep up the good works ! Your explanation is excellent and the visualization provided is highly effective!

[@PenandPaperScience]

@@Rey-zb8el Good to hear! And thank you for the kind words (:

[@dter706]

Ik kon direct horen dat je een belg was haha, moest dit even opfrissen voor bachelorproject robotica

[@PenandPaperScience]

Succes! :D