Mathematical Physics Vol 1

1.4 Operations on vectors

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whence it follows that between four vectors a , b , c and d there is always a nontrivial relation of the form α a + β b + γ c + δ d = 0 . (1.34) Thus, the relation (1.33), for each set of real numbers m , n , and p , determines a three dimensional linear vector space. One can imagine that the end point of vector d is "overwriting" all points of the 3–D space, when the parameters m , n , and p are taking all possible values from the set of real numbers. This means that in a 3–D linear vector space, each set of four vectors is linearly dependent. We will use this relation between the number of linearly independent vectors and the dimension of a space to introduce the concept of dimensionality of a three-dimensional linear vector space, noting that the concept can easily be generalized to an n –dimensional vector space. The vectors a , b and c , in (1.32) are called base vectors , and the elements of the sum m a , n b and p c components of the vector d . Numbers m , n and p will shortly be called coordinates 15 with respect to base vectors a , b and c . Once a set of base vectors is determined, then each vector is uniquely determined by a triple (in 3–D) of coordinates. A set of three mutually orthogonal vectors in 3–D space is linearly independent 16 . If orthogonal unit vectors e 1 , e 2 and e 3 are chosen as the base vectors, then each (subsequent) vector, e.g. x , can be represented by the relation x = x 1 e 1 + x 2 e 2 + x 3 e 3 . (1.35) A point in 3–D space is a geometric object (does not depend on the coordinate system). If we introduce a coordinate system, we can uniquely determine each point by an ordered triple of numbers ( x 1 , x 2 , x 3 ) , whose elements are called vector coordinates (hereafter shortly coordinates ) of x . It is said that the vectors e i , i =1,2,3, form a base or coordinate system (Fig. 1.18). The vectors ( e i ) are called (as already mentioned) base vectors.

x 3

The end points E i of the base vectors e i ( i = 1 , 2 , 3) have the following coordinates:

E 3

e 3

E 2

e 1

E 1

E 1 : E 2 :

( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) .

x 2

e 2

x 1

(1.36)

E 3 :

Figure 1.18: Base vectors and their coordinates.

Namely, vectors have previously been defined geometrically, using the oriented segment. By introducing the coordinate system, the vector can be described algebraically. It has already been 15 Note that in spaces where a scalar product is not defined, such as an affine space, there is no point in considering concepts that are defined using this product, such as magnitude or angle between two vectors. It is common in the literature that these variables, which we have called coordinates, are also called affine coordinates, thus emphasizing the nature of this (affine) space. 16 Observe a set of three mutually orthogonal vectors, for which a i · a j = A i j δ i j , where A i j = | a i |·| a j | . Let us assume that the linear combination of these vectors ∑ 3 i = 1 λ i a i = 0. By a scalar multiplication of the last relation with a j ( j = 1 , 2 , 3), and taking into account the condition of orthogonality, we obtain

3 ∑ i = 1

3 ∑ i = 1

3 ∑ i = 1 λ i A i j δ i j = λ j A j j = 0 ⇒ λ j = 0 ,

λ i a i · a j =

λ i ( a i · a j )=

which is the condition for linear independence of the observed vectors.