Mathematical Physics Vol 1

Chapter 3. Examples

58

R Note that we have used (and we will use in the future) the symbol | a | = a to denote the magnitude of the vector a (similarly for other vectors).

Problem5

Prove a) that the vector product is not commutative

a × b = − b × a , b) distributivity of the vector product with respect to vector addition a × ( b + c )=( a × b )+( a × c ) , c) distributivity of the vector product with respect to vector addition ( a + b ) × c =( a × c )+( b × c ) , d) that the following relation is true a × ( b × c )=( a · c ) b − ( a · b ) c , e) a × ( b × c )̸=( a × b ) × c .

Proof a) The magnitude of the vector a × b = c is ab sin θ and its direction is determined by the right oriented system (Fig. 3.5). The magnitude of the vector b × a = d is b · a sin θ and its direction is determined by the right oriented system (Fig. 3.5) b), ie. d = b × a = − a × b = − c .

Figure 3.5: Two possible orientations.

Given that vectors c and d have the same magnitude but opposite directions it follows that c = − d that is a × b = − b × a . Thus, the vector product is not a commutative operation. b) Let us first prove the distributivity for the case when a is normal to the plane formed by vectors b and c (Fig. 3.6).