Circles and Spheres via Dot Products - Part 1

Explore the fundamental concepts of circles and spheres through the lens of linear algebra in this 42-minute lecture. Discover two distinct approaches to introducing circles using the Euclidean dot product: the quadratic equation v.v=k and the less familiar but more powerful bilinear equation v_1.v_2=k. Delve into the applications of these methods in understanding tangents and Apollonius' theory of pole/polar duality. Examine both the unit circle and the intriguing imaginary unit circle, uncovering their significant roles in geometry. Conclude with an exploration of classical theorems about circles that form the foundation of projective geometry, gaining valuable insights into the intersection of linear algebra and geometric principles.