Amazing Things You Can Do in Geometric Algebra - Explained

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Timestamps

0:00 Vectors and scalars
1:30 Scalar-vector multiplication and vector addition
3:56 Dot product
5:21 Wedge product
8:14 Geometric product

— DISCLAIMER —

This video is intended for entertainment and educational purposes only. It should not be your sole source of information. Some details may be oversimplified or inaccurate. My goal is to spark your curiosity and encourage you to conduct your own research on these topics.

[newwaveinfantry]

I'm so glad I discovered this channel. Thanks for explaining and popularizing so many concepts in mathematics!

[TheSabian]

Ah yes, Geometric Algebra. Not to be confused with Algebraic Geometry.

[alec_smd]

Vector! with both direction! and magnitude! OH YEAHH!! - coincidentally my fav 'despicable me' quote <3

[PiboytheThird]

There are generalizations of the cross product to n dimensions.

[josegers]

We (Flanders math education) differentiate the direction of a vector. Direction only states the parallelism of vectors or line segments. We use 'sense' to define the the way the arrow points. Very confusing for students. I think we do this so that the direction of a line is the same as the the slope of that line in analytic geometry.

[linuxp]

1D and 3D are the unique meaningful physical spaces closed for multiplication of complexes, in 1D the vector product is an imaginary number and in 3D it's a set of orthogonal planes akin to imaginary components.

The whole set of G³ (polynomials of up to three vectors) is the product of vectors given by:

(x_i)² = {xx, yy, zz} = {x•x, y•y, z•z} = t

x_i = {x, y, z} = v

x_i*x_j = {yz, zx, xy} = {i(y×z), i(z×x), i(x×y)} = {iz, ix, iy} = iv

x_i*x_j*x_k = xyz = it

The real scalar (dot product) + 3 imaginary vectors (bivectors) form an object called quaternion

The set of imaginary scalar (trivector) + 3 vectors is a so called quadvectors, while the quaternions are another type of quadvectors

[WhyneedanAlias]

Idk about you but we've studied the wedge product in linear algebra.
Also we've never even touched the cross product I think

[caspermadlener]

I have some nit-picky criticism
1. The standard definition of a vector is a bit questionable, since direction doesn't really mean anything if you think about it. You can define angle though, which is more meaningful, but the axioms of vectors only require addition and scaling.
2. Your explanation of the dot product implies that it would always be positive; this is not the case.
3. I wouldn't say the wedge product is exclusive to geometric algebra, it was around way before this, and a necessary tool in representation theory, just as the general tensor product.
But it is completely true that this is the correct version of the cross product! Down with the cross product! Up with the wedge!

[Skpr]

A bivector - a vector that is bi with an attitude😅😂

[Yusdll]

It was so fun to watch this, thanks ! 🫶🏻