Mathematical Physics Vol 1

1.4 Operations on vectors

19

Definition Zero vector is a vector with zero displacement (a vector whose beginning and end coincide), and we denote it by 0 . For each vector a a + 0 = 0 + a = a . (1.7)

The magnitude of zero vector is equal to zero and its direction is arbitrary (indefinite).

Definition Two vectors of the same magnitude but opposite directions are called opposite vectors. The opposite vector to vector a is denoted by – a . For these to vectors, the following applies a +( − a )= 0 . (1.8)

Definition Each vector with a magnitude equal to one, i.e. | a | = 1

(1.9)

is called unit vector .

Based on the geometric properties of oriented segments, we conclude that: a + b = b + a ( commutativity )

(I)

( a + b )+ c = a +( b + c ) ( associativity ) (II) Also note that the vector addition operation (+) is an internal 7 binary operation, i.e.: if a , b ∈ V then also a + b ∈ V , where V is a vector set . (III) Based on the previous definitions and properties, it can be briefly summarized that the following is true for the vector addition operation: a) the operation is commutative (I), b) the operation is associative (II), c) the operation is internal (III), d) the operation has a zero (neutral) element, 0 ∈ V (1.7), e) each element a ∈ V has an opposite or symmetrical element – a ∈ V forwhich a +(- a )=(- a )+ a = 0 . (1.10) Aset V , the elements of which have properties a) to e) in relation to an operation, is said to form a commutative or Abelian 8 group, or in other words, the set V has the structure of a commutative or Abelian group. Thus, based on the previous definition, it can be said that the vector set V forms a commutative or Abelian group with respect to addition. Let us now define some more operations with vectors. 7 An internal operation assigns to each element of a set an element from the same set. 8 Niels Henrik Abel (1802-1829), Norwegian mathematician. He was the first to complete the proof demonstrating the impossibility of solving the general quintic equation in radicals. He also greatly contributed to the theory of elliptic functions and the theory of infinite series. He laid the foundation for the general theory of Abel integrals.