Definition:Dot Product

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


 * $\mathbf a = \displaystyle \sum_{k \mathop = 1}^n a_k \mathbf e_k$
 * $\mathbf b = \displaystyle \sum_{k \mathop = 1}^n b_k \mathbf e_k$

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

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

Also known as
The dot product is also known as:


 * The scalar product (but this can be confused with multiplication by a scalar so is less recommended)
 * The standard inner product.

The symbol used for the dot is variously presented; another version is $\mathbf a \bullet \mathbf b$, which can be preferred if there is ambiguity between the dot product and standard multiplication.

In the complex plane, where it is commonplace to use $\cdot$ to denote complex multiplication, the symbol $\circ$ is often used to denote the dot product.

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