Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 366

If the three vectors a , b , and c coincide with the three nonparallel edges of a
parallelepiped, the scalar triple product a is equal to the volume of the

parallelepiped. In the following example a useful vector identity for the triple vector
product ( r

ð b c Þ

( p q )) is derived.

Example A.8.4

Prove that ( r

p ) q .
Solution: First rewrite (A.114) with the change a ! r , and again with the changes
a ! p and b ! q , where b ¼

( p

q )) ¼

( r

q ) p

( r

( p q )

r b ¼ e ijk r i b j e k ; b ¼ p q ¼ e mnj p m q n e j ;

Note that the second of these formulas gives the components of b as

b j ¼ e mnj p m q n :

This formula for the components of b is then substituted into the expression for

( r b ) ¼ ( r ( p q )) above, thus

r ð p q Þ¼ e ijk e mnj r i p m q n e k :

On the right-hand side of this expression for r ð p q Þ , e ijk is now changed to

e ikj and the first of (A.111) is then employed,

r ð p q Þ¼ðd im d kn d in d km Þ r i p m q n e k :

then summing over k and i

r ð p q Þ¼ r i p k q i e k r i p i q k e k ¼ð r q Þ p ð r p Þ q :

□

In the process of calculating area changes on surfaces of objects undergoing
large deformations, like rubber or soft tissue, certain identities that involve both
vectors and matrices are useful. Two of these identities are derived in the following
two examples.

Example A.8.5
Prove that A

a

( A

b A c ) ¼ a ·( b c ) Det A where A is a 3 by 3 matrix and a , b ,

and c are vectors.

Solution: Noting the formula for the scalar triple product as a determinant

a 1 a 2 a 3

a ð b c Þ¼ b 1 b 2 b 3

c 1 c 2 c 3