Page 55 - Dynamics of Mechanical Systems

0593_C02_fm Page 36 Monday, May 6, 2002 1:46 PM

36 Dynamics of Mechanical Systems

the algorithms of Eqs. (2.8.1) and (2.8.2), we see that the vector triple products may be
expressed as:

A ×( B C) = ( A C B ) −( A B C ) (2.8.12)
⋅
⋅
⋅
and
( AB) × C = ( A C B ) −( B C A ) (2.8.13)
⋅
×
⋅
Observe that the last terms in these expressions are different.

Example 2.8.2: Validity of Eqs. (2.8.6) and (2.8.7) and the Necessity of
Parentheses
Verify Eqs. (2.8.6) and (2.8.7) using the vectors of Eq. (2.8.7).
Solution: From Eqs. (2.8.2) and (2.8.9), A × (B × C) is:

n 1 n 2 n 3
A ×( B C) = 3 −1 1 = −27 n − 29 n + 52 n 3 (2.8.14)
×
1
2
1 17 10
From Eq. (2.6.22), (A · C)B – (A · B)C is:
( AC B ) −( A C C ) = () − ( ) +− ( )( ) + () − ( )](2 n + 4 n − 7 n )
⋅
⋅
5
[ 3
1
1
1 3
1
2
3
− ()() +− ( )( ) + () − ( )] − ( n + 3 n − 5 n )
[ 32
1 4
1
7
3
2
1
)( n )
=− ( 11 2 n + 4 n − 7 (2.8.15)
1 2 3
−− ( ) − ( n + 3n − 5n )
5
3
1 2 3
=− 27n − 29n + 52n
1 2 3
Similarly, (A × B) × C and (A · C)B – (B · C)A are:
n 1 n 2 n 3
( AB) × n +
×
14
C = 3
23
=−157
2
1 n + 32 n 3 (2.8.16)
−1 3 −5
and
( AC B ) −( B C A ) = () − ( ) +− ( )( ) + () − ( )](2 n + 4 n − 7 n )
⋅
⋅
5
1
1
[ 3
1 3
3
1
2
5
− () − ( ) + ()( ) +− ( ) − ( )](3 n − n + )
n
[ 2
1
7
4 3
3
1
2
)( n )
=− ( 11 2 n + 4 n − 7 (2.8.17)
1 2 3
45
−( ) 3 ( n − n + )
n
n
1 2 3
=− 157 n + n + 32 n
1 2 3