Definition:Dot Product

Definition
Given any two vectors $a$ and $b$ in $\R^n$, the Dot Product is defined as:


 * $\displaystyle \vec a \bullet \vec b = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i-1}^{n} a_i b_i$

The symbol used for the dot is variously presented; another version is $\vec a \cdot \vec b$.

If the vectors are represented as column matrices:
 * $\vec a = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix}, \vec b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$

we can express the dot product as:
 * $\vec a \bullet \vec b = \vec a^T \vec b$

where:
 * $\vec a^T = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix}$ is the transpose of $\vec a$;
 * the operation between the matrices is the matrix product.

It is also known as:


 * The Scalar Product;
 * The Standard Inner Product.

Some basic properties of the dot product can be found here.

It can be shown that the dot product is an inner product.