Dot product: Component vs. Geometric definition
This post explains why the component definition (sum of products of corresponding coordinates) and the geometric definition (product of magnitudes times the cosine of the angle) of the dot product in Euclidean space are equivalent.
Background
Eli Bendersky is a well-known software engineer and blogger who writes clear, technical deep-dives on math, compilers, and programming. This post addresses a classic question from linear algebra: why the dot product's two definitions — the component-wise sum-of-products (algebraic) and the geometric one involving magnitudes and the cosine of the angle between vectors — are equivalent. Most textbooks present both without proving their equivalence, which can be confusing for students. The proof typically relies on the Law of Cosines or coordinate invariance in Euclidean space.