Geometrical properties of vectors

Every n vector can be represented as a point in an -dimensional coordinate space. The n elements of the vector are the coordinates along n basis vectors, such as defined in the previous section. The null vector 0 defines the origin of the coordinate space. Note that the origin together with an endpoint define a directed line segment or axis, which also represents a vector. Hence, there is an equivalence between points and axes, which can both be thought as geometrical representations of vectors in coordinate space. (The concepts discussed here are extensions of those covered previously in Sections 9.2.4 to 9.2.5.) 

In this and subsequent sections we will make frequent use of the scalar product (also called inner product) between two vectors x and y with the same dimension n, which is defined by  

Note that in Section 9.2.23 the dot product x y is used as an equivalent notation for the scalar product x y. 

In Euclidean space we define squared distance from the origin of a point x by means of the scalar product of x with itself  

It may be noted that some authors define the norm of a vector x as the square of 11 xl I rather than 11 xl I itself, e.g. Gantmacher [2]. 

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The geometric properties of vectors can be combined into the triangle relationship, also called the cosine rule, which states that ... 

Viewing concentrations as coordinates in concentration space is useful in representing the state of a system. Moreover, when this coordinate is interpreted as a vector with a unique magnitude and direction, the geometric properties of vectors may be exploited. This representation will prove to be highly useful in AR theory. Coordinate values will be enclosed in square brackets (in plaee of parentheses) to indicate veetors. 

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