Dot product: Component vs. Geometric definition
a day ago
- The dot product has two equivalent definitions: component-wise sum of products and geometric product of magnitudes times cosine of angle.
- Geometric proof uses the law of cosines on the triangle formed by vectors and their difference, leveraging distributivity.
- Projection proof assumes geometric definition and derives component form by expressing vectors in orthonormal basis and using projections.
- Appendix defines inner product space, shows the component definition satisfies inner product axioms, and discusses norm.