VECTOR ALGEBRA
θa×babFigure 7.9 The vector product. The vectorsa,banda×bform a right-handed
set.modate this extension the commutation property (7.19) must be modified to
read
a·b=(b·a)∗. (7.22)In particular it should be noted that (λa)·b=λ∗a·b,whereasa·(λb)=λa·b.
However, the magnitude of a complex vector is still given by|a|=
√
a·a,sincea·ais always real.
7.6.2 Vector productThe vector product (or cross product) of two vectorsaandbis denoted bya×b
and is defined to be a vector of magnitude|a||b|sinθin a direction perpendicular
to bothaandb;
|a×b|=|a||b|sinθ.The direction is found by ‘rotating’aintobthrough the smallest possible angle.
The sense of rotation is that of a right-handed screw that moves forward in the
directiona×b(see figure 7.9). Again,θis the angle between the two vectors
placed ‘tail to tail’ or ‘head to head’. With this definitiona,banda×bform a
right-handed set. A more directly usable description of the relative directions in
a vector product is provided by a right hand whose first two fingers and thumb
are held to be as nearly mutually perpendicular as possible. If the first finger is
pointed in the direction of the first vector and the second finger in the direction
of the second vector, then the thumb gives the direction of the vector product.
The vector product is distributive over addition, butanticommutativeandnon-associative:
(a+b)×c=(a×c)+(b×c), (7.23)b×a=−(a×b), (7.24)(a×b)×c=a×(b×c). (7.25)