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Vector SpacesAn 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
These are variants of distributive and associative laws. As, an example, for n-tuple spaces, the multiplication by a scalar is defined componentwise:
With this definition and addition defined also componentwise the set of 3-tuples becomes a vector space. It's important to understand that an n-tuple 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 ProductFor 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:
As an example, the scalar product for 3-tuples is defined in the following manner:
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 3-tuples are orthogonal:
a and
If we introduce the length (also called the norm) of vector a as
Identity (*) is a generalization of the Cosine Law. In fact (*) is one of the reasons that the angle between two vectors is defined by:
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