Mathematical Physics Vol 1

1.4 Operations on vectors

25

(a)

(b)

(c)

Figure 1.12: Right-screw rule (a), and right-hand rule (c)

These conditions uniquely determine the vector c . The vector product is symbolically denoted by:

a × b = c ,

(1.20)

which is read as "a cross b". In mechanics (physics) the vector product has the following physical meaning. Consider rotating a body around a fixed point. This rotation is due to the action of moment. The moment of force S for a point is defined by the following relation O = r × S , where r is the position vector of the point of application of the force relative to the moment point O . Note that the following holds for the vector product: - it is distributive with respect to addition: M S

a × ( b + c )=( a × b )+( a × c ) ( a + b ) × c =( a × c )+( b × c )

(1.21) (1.22)

- it is not commutative , as (Fig. 1.13)

a × b = − b × a (anticommutativity)

(1.23)

- it is not associative , as in general

a × ( b × c )̸=( a × b ) × c .

(1.24)