Let's Invent Two-Dimensional Multiplication

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An intuitive derivation of the geometric definition of the dot product, a multiplicative operation between two elements of the cartesian plane R^2. We'll consider how multiplication of real numbers works, and how the many directions between vectors in R^2 can be translated into a meaningful factor in our operation. We'll then see the component definition of the dot product, and prove it is equivalent to the geometric one using the law of cosines. We'll complete several examples and finish by discussing angles and dot products in four dimensions. #mathematics #multivariablecalculus #maths

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0:00 Intro to 2D Space
1:10 Our Objectives
1:27 Creating the Dot Product
10:17 The Cosine Function
14:07 Defintion
14:53 Radians
15:47 Trying the Dot Product
18:23 An Easier Way
20:43 Vector Operations
22:35 Law of Cosines
24:19 Self Dot
25:32 Proof of Equivalence
29:01 Finding Angles
31:20 Angle in 4D
33:54 Conclusion