Mathematical Physics Vol 1

1.4 Operations on vectors

23

- the scalar product of an arbitrary vector with itself is non-negative

2

a · a = | a |

> 0 , and

(VII)

a · a = 0 , if a = 0 , (positively – definite)

- the scalar product is commutative

a · b = b · a , (symmetry)

(VIII)

- the scalar product is distributive with respect to addition

a · ( b + c )= a · b + a · c ,

(IX)

- the scalar product is associative with respect to multiplication by a scalar α ( a · b )=( α a ) · b = a · ( α b ) , where α is a real number . Some other properties that follow from the definition of a scalar product are:

(X)

| a · b |≤| a |·| b | ,

(1.15)

(Schwarz inequality) 10

| a + b |≤| a | + | b | , (triangle inequality)

(1.16)

2

2

2

2

| a + b |

+ | a − b |

= 2 ( | a |

+ | b |

(1.17)

) .

(parallelogram equality)

A real affine space V or real vector space in which the scalar product of a vector with properties (VII)–(X) is defined, is called the real Euclidean 11 space . The concept of Euclidean space defined in this way is used to define a more general concept of Euclidean space. A set E , with elements of an arbitrary nature, for which the following is axiomatically defined: 1) an addition operation with properties (I)–(III), 2) an multiplication operation of elements of set E by elements of a field R , with properties (IV)–(VI) and 3) a multiplication operation with properties (VII)–(X), is called Euclidean space over the field R . Let us now define an orthonormal set of vectors. 10 Hermann Amandus Schwarz (1843-1921), German mathematician, known for his work in complex analysis (conformal mapping), differential geometry and calculus of variations. 11 E υκλειδης , born about 330 BC, and died about 275 BC. One of the greatest Greek mathematicians of the ancient era. He was one of the founders and central figure of the mathematics school in Alexandria. He has written several works on geometry, optics and astronomy. His most important work is Elements ( Σ τ o ιχε ˜ ια ) .