The Geometry Behind the Dot Product: Unit Vectors, Projections, and Intuition

A foundational guide to the geometric concepts underlying the dot product. Covers unit vectors and normalization, scalar projection (with the shadow analogy), vector projection (with a trail analogy), and derives the dot product formula geometrically using a rotation trick. Explains how the dot product of two unit vectors equals the cosine of the angle between them, and how scalar and vector projections relate to each other. Part one of a three-part series leading toward applications in recommendation systems and NLP.