Rotation Matrix for Coordinate Transformation

Physics Ninja looks at the simple proof of calculating the rotation matrix for a coordinate transformation. The transformation is used to write the components of a vector or the coordinates of a point in terms of a local coordinate that is rotated by some angle relative to another.

[arryolo]

Thank for the help. Was initially worried how complicated it look when I looked at my textbook, but it was WAY simpler when you explained it.

[rugoodbro]

I literally spent the past couple of hours tryna understand this, watched too many videos but still couldn't understand and then I found your video and understood it! thank u so much

[raylittlerock]

Clear and efficient explanation, the construction proposed is much simpler than others I've seen elsewhere.

[isagumus]

Swear I was trying to understand this during whole my bachelor of engineering but you sir, taught me that in only 503 seconds!

[Matt-rkjk]

This is great - thanks a bunch! I am currently in a FEA class, and had to quickly brush up on some matrix transformations between coordinate systems, and this is the exact video I needed.

[abmj]

Love you I wasted whole day on this topic and you have done it in less than 10 min❤

[xithcal]

Great explanation, wasted hours trying to figure it out and you helped me understand it in less than 5 minutes :D. Thanks!

[chadify]

Great work, how would this go down if someone were to have to determine the rotation matrix instead and were given A (as a set of coordinates). And B as the vertical alignment of those coordinates as in Z = sqrt (x^2 + y^2) being the y component of the resultant B factor

[johngrant]

This would be much more helpful if I could see what you mean when you say "this one" frequently.

[physicsstudent]

Very good explanation! Finally I understood this content. Thank you!

[Helmutandmoshe]

It would be really great and helpful to many, many students, if you were to organize your videos into playlists!

[waleedal-zaidi]

thank you so much, you saved the day ;)

[UnstopableWords]

I was totally confused, and thinking I could not understand the transformation of the coordinate system even though I spent hours on chatgpt and Gemini but both could not teach me this concept. This man more intelligent 🧠 then AI 😅
Before watching this video I decided that if I grasp this concept from this video i will like the video then subscribe to the channel and express my feelings in the comment section.
Thank you.

[isaacrodriguez-padilla]

According to your diagram, the rotation matrix you're getting is for a counter-clockwise rotation. Why does wikipedia states that your rotation matrix corresponds to a clockwise rotation?

[kalumbabwale]

I'm distracted by the chaos on the bookshelf 😄

[spurti]

Hii could you make a video on phasors??

[MohamedEzzat]

Rθ = [cosθ -sinθ]
[sinθ cosθ]
i found it in many sources what is the difference !?

[mariostelzner]

YEAH! THIS MADE ABOUT MUCH SENSE AS THE PROBLEM THAT SENT ME HERE. THANKS EINSTEIN. AHAHAHA AHAHAHA LOL

[ndrklerz]

That is simple and ideas are clear but it is not a proof because you are using one drawing which is just a special case. You need to prove that for point being anywhere in coordinate system and for angle of rotation being anything. Notice your vector has angle of about 45 degrees and your rotation angle is about 30 degree. *So that is what your "proof" is good for*. I mean this is OK to explain to concepts but it is inadequate for mathematics. It is sloppy and it obfuscates the notion of proof. Not to mention that you actually used quite a few Euclidean Geometry propositions pretending that they just hold ("you should convince yourself"). In other words this is just a sloppy mathematics. The reality is that people that write math textbook cannot write this type of sloppy explanations because they have obligation to write a complete proof. If you really want you can turn the idea this man is using into a complete proof with some trigonometry but then it would look quite complicated and certainly worse than a classic proof using polar coordinates. Yes, mathematics does not allow sloppiness and mathematicians do not call special cases a "proof."