Mathematical Physics Vol 1

Chapter 1. Vector algebra

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Definition It is said that a set of three vectors (in 3-D Euclidean space) e 1 , e 2 , e 3 is an orthogonal normalized set or shortly orthonormal set , if the following condition is satisfied: e i · e j = δ i j = 1 , i = j 0 , i̸ = j i , j = 1 , 2 , 3 . (1.18)

The previous definition also applies in the n -dimensional Euclidean space E n , where the indices i and j , in relation (1.18), take the values i , j = 1 , 2 ,..., n .

The variable δ i j , defined by the previous relation, is referred to in the literature as Kro necker’s 12 delta symbol.

1.4.5 Vector (cross) product of two vectors

Definition A vector product of two vectors a and b in E 3 is avector c determined by the following conditions: i ) c is perpendicular to both a and b , and thus normal to the plane containing vectors a and b ; ii ) direction of the vector c is given by the right-hand rule (or the right-screw rule). Namely, if we point the thumb of our right hand in the direction of vector a , and our index finger in the direction of vector b , and then rotate the vector a by a sharp angle (in the positive direction) to coincide with vector b , then the tip of the middle finger will indicate the direction of the vector product (see figures 1.12a, 1.12b and 1.12c); iii ) magnitude of the vector c is determined by the relation: | c | = | a |·| b |· sin α , α = ∠ ( a , b ) . (1.19)

12 Leopold Kronecker (1823-1891), German mathematician, who gave a significant contribution to algebra, group theory and number theory.