Explore advanced applications of the dot product in planar geometry in this 46-minute lecture. Delve into linear functionals, point-normal form of lines, and normal vectors. Learn to derive the formula for vector projection and calculate the quadrance of a point to a line. Examine isometries, including reflection in a line perpendicular to a vector, and discover how the product of two reflections results in rotation. Gain a deeper understanding of these fundamental concepts in linear algebra and their practical applications in geometric analysis.