Let's Invent Two-Dimensional Multiplication

Join the channel to get exclusive and early videos, original music, lecture videos, and more!    / @wrathofmath   🛍 Check out my math fashion brand: https://mathshion.com/ More Math Chats:    • Math Chats   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 Textbooks I Like! Graph Theory: https://amzn.to/3JHQtZj Real Analysis: https://amzn.to/3CMdgjI Abstract Algebra: https://amzn.to/3IjoZaO Linear Algebra: https://amzn.to/43xAWEz Calculus: https://amzn.to/3PieD1M Proofs and Set Theory: https://amzn.to/367VBXP (available for free online) Statistics: https://amzn.to/4d5HbUQ Discrete Math: https://amzn.to/3qfhoUn Number Theory: https://amzn.to/3JqpOQd ★DONATE★ ◆ Support Wrath of Math on Patreon:   / wrathofmathlessons   ◆ Donate on PayPal: https://www.paypal.me/wrathofmath Follow Wrath of Math on... ● Instagram:   / wrathofmathedu   ● TikTok:   / wrathofmathedu   ● X: https://x.com/wrathofmathedu 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