Vector Space and Spaces with the Scalar Products
An abstract real vector space is an commutative group with
one additional operation: its elements may be multiplied by real numbers (scalars). It's by no means a
group operation (except for the case when we look at the set R of real numbers as a real vector space)
because in a group operations both operands must come from the same set. Multiplication by a scalar
is required to satisfy three additional laws: for
 (distributivity): (u + v)a = ua + va
 (associativity): u(va) = (uv)a
 (distributivity): u(a + b) = ua + ub
These are variants of distributive and associative laws. As, an example, for ntuple spaces, the multiplication by a scalar is defined componentwise:
u(a_{1}, a_{2}, a_{3}) = (ua_{1}, ua_{2}, ua_{2}) 
With this definition and addition defined also componentwise the set of 3tuples becomes a vector space. It's important to understand that an ntuple is only then is regarded as a vector when it's considered an element of a set where two operations (addition and multiplication by a scalar) are defined. Thus, vectors and Vector Spaces are born simultaneously.
Spaces with Scalar Product
For some vector spaces it's possible to define another multiplication  a scalar (or inner, or dot) product. The scalar product is defined for two vector operands with the result being a scalar. Therefore, the scalar product too is not a group operation. The scalar product of two vectors a and b is denoted a.b or (a, b) and has the following properties:
 (commutativity): a.b = b.a
 (distributivity): a.(b + c) = a.b + a.c
As an example, the scalar product for 3tuples is defined in the following manner:
(a_{1}, a_{2}, a_{3}).(b_{1}, b_{2}, b_{3}) = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} 
As an application of these laws, let's prove a simple but interesting identity.
(*) 

Two vectors whose scalar product is zero are called orthogonal or perpendicular. For example, the following pairs of 3tuples are orthogonal:
a and
(a + b).(a + b) = a.a + b.b 
If we introduce the length (also called the norm) of vector a as
a  b^{2} = a^{2} + b^{2}. 
Identity (*) is a generalization of the Cosine Law. In fact (*) is one of the reasons that the angle between two vectors is defined by:
cos(α) = a.b / a b. 
What Can Be Multiplied?
 What Is Multiplication?
 Multiplication of Equations
 Multiplication of Functions
 Multiplication of Matrices
 Multiplication of Numbers
 Peg Solitaire and Group Theory
 Multiplication of Permutations
 Multiplication of Sets
 Multiplication of Vectors
 Multiplication of a Vector by a Matrix
 Vector Space and Spaces with the Scalar Product
 Addition and Multiplication Tables in Various Bases
 Multiplication of Points on a Circle
 Multiplication of Points on an Ellipse
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