Mathematical Methods for Physics and Engineering - Matematica.NET

VECTOR ALGEBRAa × bθbaFigure 7.9set.The vector product. The vectors a, b and a×b form a right-handedmodate this extension the commutation property (7.19) must be modified toreada · b =(b · a) ∗ . (7.22)In particular it should be noted that (λa) · b = λ ∗ a · b, whereas a · (λb) =λa · b.However, the magnitude of a complex vector is still given by |a| = √ a · a, sincea · a is always real.7.6.2 Vector productThe vector product (or cross product) of two vectors a and b is denoted by a × band is defined to be a vector of magnitude |a||b| sin θ in a direction perpendicularto both a and b;|a × b| = |a||b| sin θ.The direction is found by ‘rotating’ a into b through the smallest possible angle.The sense of rotation is that of a right-handed screw that moves forward in thedirection a × b (see figure 7.9). Again, θ is the angle between the two vectorsplaced ‘tail to tail’ or ‘head to head’. With this definition a, b and a × b form aright-handed set. A more directly usable description of the relative directions ina vector product is provided by a right hand whose first two fingers and thumbare held to be as nearly mutually perpendicular as possible. If the first finger ispointed in the direction of the first vector and the second finger in the directionof the second vector, then the thumb gives the direction of the vector product.The vector product is distributive over addition, but anticommutative and nonassociative:(a + b) × c =(a × c)+(b × c), (7.23)b × a = −(a × b), (7.24)(a × b) × c ≠ a × (b × c). (7.25)222