This is the most important video in this series---and is possibly the most important mathematics video you are likely to see. It introduces Rational Trigonometry from first principles using a vector approach. The main notions of quadrance and spread replace distance and angle, and are introduced purely algebraically. The scalar/inner/dot product plays an important role, and allows us to introduce perpendicularity algebraically, and to introduce the spread between two vectors. The main laws are Pythagoras' theorem, the Triple quad formula, the Cross law, the Spread law and the Triple spread formula. The proofs of the main laws are given as Exercises, but I give some hints. Please spend time to write out solutions carefully!For more information on this purely algebraic approach to trigonometry and geometry at an elementary level, see my YouTube playlist WildTrig, and my book'Divine Proportions: Rational Trigonometry to Universal Geometry'.
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