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. 
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Copyright © 19962018 Alexander Bogomolny