Page 71 - Dynamics of Mechanical Systems

0593_C02_fm Page 52 Monday, May 6, 2002 1:46 PM

52 Dynamics of Mechanical Systems

a. Find a unit vector n parallel to L, in the direction of P to Q. Express the results
in terms of the unit vectors n , n , and n shown in Figure P2.7.7.
z
y
x
b. Express F in terms of n , n , and n .
z
y
x
c. Form the vectors OP and OQ, calculate OP × F and OQ × F, and express the
results in terms of n , n , and n . Compare the results.
z
y
x
Z
F = 7 lb
P (1,-1,3)
F
n
z
Q (2,4,1)
L
Y
O
n
FIGURE P2.7.8 y
Coordinate system X, Y, Z with points P n
and Q, line L, and force F. X x
P2.7.9: Let e and δ be the permutation symbol and the Kronecker delta symbol as in
jk
ijk
Eqs. (2.7.7) and (2.6.7). Evaluate the sums:
3
3
∑ ∑ e δ = 12 3
,,
i
ijk
jk
j=1 k=1
Section 2.8 Vector Multiplication: Triple Products
P2.8.1: See Example 2.8.1. Verify the remaining equalities of Eq. (2.8.5) for the vectors A,
B, and C of Eq. (2.8.7).
P2.8.2: Use Eq. (2.8.6) to find the volume of the parallelepiped shown in Figure P2.8.2,
where the coordinates are measured in meters.

Z
k
C(1,1,5)

O Y

A(6,0,0) j
B(2,8,0)
FIGURE P2.8.2 i
Parallelepiped. X
P2.8.3: The triple scalar product is useful in determining the distance d between two non-
parallel, non-intersecting lines. Specifically, d is the projection of a vector P P , which
1 2
connects any point P on one of the lines with any point P on the other line, onto the
1
2
common perpendicular between the lines. Thus, if n is a unit vector parallel to the common
perpendicular, d is given by:
⋅
d = PP n
12