Mathematical Physics Vol 1

Chapter 3. Examples

66

Proof The condition is necessary . If a , b and c are in the same plane, then the volume of the parallelepiped they form is equal to zero. From there, it follows that a · b × c = 0 (see example 11 on p. 63). The condition is sufficient . If a · b × c = 0 then the volume of the parallelepiped formed by these three vectors is equal to zero, and thus it follows that they lie in the same plane, i.e. they are coplanar.

Problem20 Let a · b × c̸ = 0 and a ′ =

c × a a · b × c

a × b a · b × c

b × c a · b × c

b ′ =

c ′ =

. Prove

a) a ′ · a = b ′ · b = c ′ · c = 1, b) a ′ · b = a ′ · c = 0 , b ′ · a = b ′ · c = 0 , c ′ · a = c ′ · b = 0, c) a ′ · b ′ × c ′ = 1 V , if a · b × c = V . d) a ′ , b ′ and c ′ are not coplanar, given that a , b and c are not coplanar (initial assumption).

Solution a)

b × c a · b × c c × a a · b × c a × b a · b × c

a · b × c a · b × c a · b × c a · b × c a · b × c a · b × c

a ′ · a = a · a ′ = a · b ′ · b = b · b ′ = b · c ′ · c = c · c ′ = c ·

= 1 .

=

= 1 .

=

= 1 .

=

b)

b × c a · b × c

b · b × c a · b × c b × c a · b × c

c · b × b a · b × c b · c × c a · b × c

a ′ · b = b · a ′ = b ·

= 0 ,

=

=

c · b × c a · b × c

a ′ · c = c · a ′ = = 0 . The same can be proved for other vector products analogously. = c · =

The same result follows from the fact that vector a ′ has the same direction as the vector product b × c and is therefore perpendicular to the plane formed by these two vectors a ′ · ( α b + β c )= 0. Note that, according to the definition (1.25), sets of vectors { a , b , c } and { a ′ , b ′ , c ′ } form a reciprocal or conjugate system of vectors. c) Given that, according to the assumption a · ( b × c )= V , it follows that a ′ · ( b ′ × c ′ )= b × c a · ( b × c ) · c × a a · ( b × c ) × a × b a · ( b × c ) = = 1 V 3 ( b × c ) · [( c × a ) × ( a × b )] .