A new three-part series on Towards Data Science breaks down the geometric foundations of the dot product, a fundamental operation in machine learning. The first installment focuses on unit vectors, scalar projections, and vector projections - concepts that are essential for understanding how the dot product works. The article explains that the dot product measures directional alignment between vectors, with values ranging from -1 (opposite directions) to 1 (same direction).

Unit vectors play a central role in this geometric interpretation. Any non-zero vector can be decomposed into its magnitude multiplied by its unit vector, which preserves direction while having length 1. The article demonstrates that the dot product of two unit vectors equals the cosine of the angle between them, providing an intuitive similarity measure. This relationship holds in any number of dimensions because any two vectors always lie in a single flat plane that can be rotated to align with the xy-plane.

The scalar projection concept is explained through a shadow analogy - how much of one vector lies along the direction of another. This is calculated as the product of one vector's magnitude and the cosine of the angle between them. The article shows that multiplying a vector by a unit vector extracts its component along that direction, making unit vectors act as direction filters. The next parts will explore the dot product's geometric meaning, relationship to cosine similarity, and applications in recommendation systems and NLP.