Geometric Algebra

Currently Playing: The polar form of the geometric product | Geometric algebra episode 5

#geometricalgebra #geometricproduct #multivectors #complexnumbers #euler The geometric product produces complex numbers, which we can write in their polar form. This finally tells us what the geometric product is good for: it produces rotations. It allows us to take any two unit vectors and construct the rotation from the first unit vector to the second one. Using Euler's formula and the Taylor series for the exponential function, we can also write this in its exponential form. The bivector can even tell us which plane we're rotating in. You can support us on Patreon, where you can already watch all the Geometric Algebra videos and get access to exclusive content. Unfortunately, we won't be publishing any new videos after Geometric Algebra, at least not in the near future. Your support is still more than welcome of course: https://www.patreon.com/user?u=86649007 I have only 1 reference for you today, but it's a really good one: [MOM 1] https://www.youtube.com/watch?v=lkhBcWtFxKU This is a very cool application of geometric algebra. It derives the law of sines & cosines in trigonometry. Even simple trigonometry can be reduced to geometric algebra! 0:00 Introduction & recap 1:02 The dot product contains a cosine 1:58 The wedge product contains a sine 2:57 The polar form of the geometric product 4:28 The geometric product produces rotations 5:02 Why the geometric product is invertible 6:45 The exponential form of the geometric product 9:42 The bivector knows the plane of rotation 10:24 The Taylor series for the exponential function 11:09 Summary This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )