sudgylacmoe | The Algebra of Vectors - From Zero to Geo 1.4 @sudgylacmoe | Uploaded October 2021 | Updated October 2024, 2 hours ago.
While we have been naming some of the operations on vectors after operations on numbers, how far do these similarities go? In this video, we go over many of the proofs of algebraic properties of vectors, showing that they do act in a similar way to numbers.

This video is a part of "From Zero to Geo", a series where we formulate geometric algebra, an incredibly powerful branch of mathematics, from the ground up. Full playlist here: youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy
Next video here: youtube.com/watch?v=O40hzr71N0w

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Sections:
00:00 Introduction
02:43 Commutativity of Addition
03:45 Associativity of Addition
05:18 Additive Identity
06:19 Additive Inverses
06:58 Length Nonnegativity
07:29 Length of Zero
07:59 Length of Negative
08:22 Length of Product
09:00 Triangle Inequality
09:50 Commutativity of Multiplication
10:29 Associativity of Multiplication
12:16 Distributivity
14:56 Multiplicative Identity
15:21 Multiplying by Zero
16:43 Multiplicative Inverses
19:23 Conclusion