Definition:Dot Product

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


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

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

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:
 * $\vec a \cdot \vec b = \vec a^T \vec b$

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.

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


 * $\vec a \cdot \vec b = \left\Vert{ \vec a }\right\Vert \left\Vert{ \vec b }\right\Vert \cos \angle \vec a, \vec b$

where:
 * $\left\Vert{ \vec a }\right\Vert$ is the length of $\vec a$ and $\left\Vert{ \vec b }\right\Vert$ is the length of $\vec b$
 * $\angle \vec a, \vec b$ is the angle between $\vec a$ and $\vec b$, taken to be between $0$ and $\pi$.

It can be shown that these two definitions are equivalent.