07-24-2026, 09:43 PM
Dot product: Component vs. Geometric definition
Summary
Eli Bendersky’s guide explores why the component definition of the vector dot product ($\vec{a}\cdot\vec{b} = \sum a_i b_i$) and its geometric definition ($\vec{a}\cdot\vec{b} = \vert{}\vec{a}\vert{}\vert{}\vec{b}\vert{}\cos\theta$) are mathematically equivalent in Euclidean space. To demonstrate this equivalence, the article outlines two distinct approaches: a geometric proof based on the Law of Cosines and a projection proof using orthonormal basis vectors. In the geometric proof, the author considers the triangle formed by vectors $\vec{a}$, $\vec{b}$, and their difference vector $\vec{c} = \vec{a} - \vec{b}$. By writing the Law of Cosines for this triangle and expanding the self-dot product $(\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})$ using the distributive property, canceling equal terms on both sides smoothly isolates $\vec{a}\cdot\vec{b}$ to yield the geometric formula.
The projection proof works in reverse by starting from the geometric definition and evaluating the dot product of a vector with an orthonormal basis vector $\vec{e}_i$. This step uses trigonometry to show that projecting a vector onto a basis vector isolates its scalar component $a_i$. Expressing both arbitrary vectors as linear combinations of basis vectors and expanding their dot product algebraically then transforms the geometric expression directly into the sum of component products $\sum a_i b_i$.
Finally, the guide's appendix provides rigorous theoretical background by showing that the component definition satisfies the formal axioms of an inner product space—namely symmetry, linearity, and positive-definiteness. This groundwork validates key properties like bilinearity and connects the vector norm directly to a generalized form of the Pythagorean theorem.
ARTICLE
Summary
Eli Bendersky’s guide explores why the component definition of the vector dot product ($\vec{a}\cdot\vec{b} = \sum a_i b_i$) and its geometric definition ($\vec{a}\cdot\vec{b} = \vert{}\vec{a}\vert{}\vert{}\vec{b}\vert{}\cos\theta$) are mathematically equivalent in Euclidean space. To demonstrate this equivalence, the article outlines two distinct approaches: a geometric proof based on the Law of Cosines and a projection proof using orthonormal basis vectors. In the geometric proof, the author considers the triangle formed by vectors $\vec{a}$, $\vec{b}$, and their difference vector $\vec{c} = \vec{a} - \vec{b}$. By writing the Law of Cosines for this triangle and expanding the self-dot product $(\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b})$ using the distributive property, canceling equal terms on both sides smoothly isolates $\vec{a}\cdot\vec{b}$ to yield the geometric formula.
The projection proof works in reverse by starting from the geometric definition and evaluating the dot product of a vector with an orthonormal basis vector $\vec{e}_i$. This step uses trigonometry to show that projecting a vector onto a basis vector isolates its scalar component $a_i$. Expressing both arbitrary vectors as linear combinations of basis vectors and expanding their dot product algebraically then transforms the geometric expression directly into the sum of component products $\sum a_i b_i$.
Finally, the guide's appendix provides rigorous theoretical background by showing that the component definition satisfies the formal axioms of an inner product space—namely symmetry, linearity, and positive-definiteness. This groundwork validates key properties like bilinearity and connects the vector norm directly to a generalized form of the Pythagorean theorem.
ARTICLE
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