Definition:Dot Product

Definition
Let $\mathbf a$ and $\mathbf b$ be vectors in a vector space $\mathbf V$ of $n$ dimensions:

where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.

Real Euclidean Space
In the context of a real Euclidean space $\R^n$, the definition can be geometrical in nature:

Complex Numbers
The definition continues to hold when the vector space under consideration is the complex plane:

Also see

 * Equivalence of Definitions of Dot Product
 * Cosine Formula for Dot Product


 * Properties of Dot Product
 * Dot Product is Inner Product


 * Definition:Vector Cross Product


 * Definition:Scalar Multiplication on Vector Space