Why can't you multiply vectors?

The speaker, Freya, an independent game developer and educator, introduces the topic of mathematics in game development. She expresses her passion for math and its deep integration into game development processes. Freya discusses her journey from an environment artist to learning programming and delving into math. She addresses the perceived divide between math and programming and her goal to bridge this gap. She also outlines her professional background, including co-founding NE Corp, developing games like Budget Cuts, and creating Unity plugins.

Freya delves into the question of why vectors can't be multiplied using traditional arithmetic, which leads to an exploration of different types of vector products, such as the dot product, cross product, and Hadamard product. She explains the concept of a canonical product and how multiplying vectors doesn't result in a simple vector output, thus complicating the idea of a straightforward multiplication.

The speaker discusses the concept of mathematical closure in different number sets, starting with natural numbers, which are not closed under subtraction, leading to the introduction of integers. She explains that integers are closed under addition, multiplication, and subtraction, but not division, which introduces rational numbers. The exploration continues with exponentiation and the introduction of real numbers and complex numbers, which are closed under all operations.

Freya draws parallels between complex numbers and vectors, noting their similar component structures. She explains how complex numbers can be visualized in a complex plane, analogous to a 2D vector space. The multiplication of complex numbers is explored, leading to an algebraic formula that results in another complex number, demonstrating the closure of complex numbers under multiplication.

The speaker attempts to multiply two vectors directly, resulting in an algebraic expression that does not simplify to a standard vector. This leads to the conclusion that vectors are not closed under multiplication. However, the exploration doesn't end there; Freya seeks to understand the resulting algebraic structure, which she likens to a quest for knowledge.

Freya introduces a 'divine axiom' that states when a vector is multiplied by itself (squared), the result is the length of the vector squared. Using this axiom, she explores the multiplication of basis vectors and discovers that certain components, like YZ, ZX, and XY, behave similarly to the cross product. This leads to the accidental discovery of the dot product and the cross product through algebraic manipulation.

The talk continues with the exploration of the algebraic constructs resulting from vector multiplication, leading to the emergence of quaternions in 3D and complex numbers in 2D. Freya explains that these constructs are not simply vectors but have their algebraic behaviors, which are foundational to understanding geometric algebra and Clifford algebra.

Freya connects the concept of curvature in mathematics and physics to the algebraic structures discussed earlier. She shows how the wedge product can simplify the understanding and calculation of curvature in various dimensions. The talk concludes with a discussion on geometric algebra, specifically Clifford algebra, and how it provides a unified framework for understanding complex numbers, quaternions, and the cross product.

The speaker shares her ongoing work with quaternionic splines in a library for Unity and her interest in exploring further mathematical concepts. She discusses the potential application of her findings in game development and her YouTube channel's educational content. Freya also addresses questions about the practical use of the concepts presented, the possibility of her tools being adapted for other engines, and her future plans.

Freya concludes the talk by thanking the audience for their attention and opens the floor for questions. She expresses her enthusiasm for sharing knowledge and invites the audience to discuss further or approach her after the talk. The speaker also hints at a potential return to game development and the possibility of creating standalone tools.