Coordinate Transformation: Rotations - Exercises

[@insertusername5737]

Finally someone that is clear to explain, it's so hard to find this quality of explanation on these topics. Thank you so much

[@PenandPaperScience]

Thank you so much for the kind words! :)

[@PenandPaperScience]

Timestamps: 0:00 Introduction 1:00 First Exercise 7:45 Second Exercise 13:40 Outro Hope it helps :)

[@abhivaryakumar3107]

Man this video is amazing I couldnt find ANYTHING good on this topic You really really helped me for my exam Thanks my man

[@PenandPaperScience]

I am glad you found this video then! Good luck witht the exam!! 🤞😎

[@chidumebichukwuneke8274]

Why did we use the negative of the angle we found in the rotational matrix

[@wolowizard4441]

was also coming to ask that

[@PenandPaperScience]

Excellent question! I try to explain this starting at 02:20 Basically, the rotation matrix is derived specifically for rotating vectors inside a fixed coordinate system. What we do here, however, is not rotate the vector, but rather the coordinate axis themselves! Now imagine that rotating a vector over an angle alpha, is the same as rotating the coordinate axis over an angle -alpha when leaving the vector unchanged. I have a video where I derive the rotation matrix, perhaps that one also helps: kzbin.info/www/bejne/e4vYl5x_rNykfKM If my explanation is not clear, feel free to follow-up (:

[@evaristensabimana1541]

Why do we use the rotation angle with opposite sign?

[@PenandPaperScience]

It all depends on the vantage point: rotating a vector over an angle alpha with the x-y axes fixed, is identical to rotating the x-y axes over an angle -alfa with the vector fixed.

[@wolowizard4441]

why can we rewrite alpha as tan(-1) ?

[@PenandPaperScience]

Good question! Using the triangle I drew at 09:45, we can use the definition of the tangent of alpha to equate it to y/x. Now, to get the angle alpha out of the tangent, we have to apply the inverse function of the tangent. This is done in the same way to get an "x" out of a square root. The inverse of the square root is taking the square. The inverse of the tangent is written as tan^-1. Does this make sense? If not, let me know.

[@slifer4963]

Hey there, in the first exercise, do you know why [x' y"] is written in a 2x1 matrix instead of a 1x2 matrix? Like : | a | | b | instead of [a b]

[@PenandPaperScience]

I'm not entirely sure, but it *has* to be written as a 2x1 matrix for the matrix multiplication to be valid. If you want to get the new vector R_new = Matrix * R_old. Does that answer your question?