Page 73 - Dynamics of Mechanical Systems

0593_C02_fm Page 54 Monday, May 6, 2002 1:46 PM

54 Dynamics of Mechanical Systems

c. Let unit vector n be expressed in terms of n , n , and n as:
z
x
y
a
n = a n + a n + a n
a x x y y z z
show that I may then be expressed as:
a
I = a I + a I + a I
a x x y y z z

Section 2.9 Use of the Index Summation Convention
P2.9.1: Evaluate and/or expand the following terms:

a. δ kk
b. e δ
ijk jk
c. a b
ij jk

Section 2.10 Review of Matrix Procedures
P2.10.1: Given the matrices A, B, and C:

1 −2  4 4 8 −  3 7 −1 −  7
     
A = 5 8 3 B = 5 −6 9 C = 3 6 8 





7 −6  9   0 5 2    4 −2 9  

a. Compute the products AB and BC.
b. Compute the products (AB)C and A(BC) and compare the results (see Eq.
(2.8.10)).
T
T
T
T
c. Find B , A , and the product B A .
T
T
d. Find (AB) and compare with B A (see Eq. (2.10.9)).
T
ˆ
e. Find A , the matrix of adjoints for A (see Eq. (2.10.11)).
f. Compute A .
–1
–1
–1
g. Compute the products A A and AA .
P2.10.2: Given the matrix S (an orthogonal matrix):
 12 6 4 6  4
 

S =− 3 2 24 24 
 
 0 − 2 2 2 2 
a. Find detS.
–1
b. Find S .
T
c. Find S .
–1
T
d. Compare S and S .