Explore the foundations of Einstein's Special Theory of Relativity through an in-depth examination of the planar relativistic dot product. Learn how this small variation on the Euclidean dot product impacts geometric concepts, including quadrance, Pythagoras' theorem, linear functionals, line equations, projections, and circles. Discover the similarities and differences between Euclidean and relativistic geometries, gaining valuable insights into the mathematical underpinnings of modern physics. Engage with exercises to reinforce understanding and conclude with a comprehensive overview of the topic's significance in both mathematics and theoretical physics.