Dot product

1. Overview

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. 

2. Description

2.1 Algebraic definition

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. 

2.2 Algebraic Notation

$$\alpha =a\cdot b=\left \langle a,b \right \rangle=a^{T}b=\sum_{i=1}^{n}a_{i}b_{i}$$

2.3 Geometric Notation

$$\alpha =cos(\theta _{ab})\left \| a \right \|\left \| b \right \|$$

2.4 Properties

2.4.1 Distributive

$$a\cdot (b+c)=a\cdot b+a\cdot c$$

2.4.2 Not associative

$$a^{T}(b^{T}c)\neq (a^{T}b)^{T}c$$

2.4.3 Commutative

$$a\cdot b=b\cdot a$$

$$a^{T}\cdot b=(a^{T}\cdot b)^{T}=b^{T}\cdot a$$

3. References

https://en.wikipedia.org/wiki/Dot_product