A mathematical explanation of why the component and geometric definitions of the vector dot product are equivalent. Two proofs are presented: a geometric proof using the law of cosines on the triangle formed by two vectors, and a projection proof that starts from the geometric definition and derives the component form using orthonormal basis vectors. An appendix covers inner product spaces, proving symmetry, linearity, and positive-definiteness for the standard dot product operation, along with the definition of vector norms.
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