454 Chapter 16Vectors
In terms of the components of the vector, ifa 1 = 1 a
xi 1 + 1 a
yj 1 + 1 a
zk, wherei,j,andk
are (constant) base vectors, the dependence ofaon tis that of the components:
a(t) 1 = 1 a
x(t)i 1 + 1 a
y(t)j 1 + 1 a
z(t)k (16.22)
and
(16.23)
EXAMPLE 16.8Findda 2 dtandd
2a 2 dt
2fora(t) 1 = 13 t
2i 1 + 121 sin 1 t 1 j 1 + 1 e
−tk.
0 Exercises 15, 16
We note that the base vectors may also depend on tif the coordinate system itself (the
frame of reference) is undergoing changes
Parametric representation of a curve
Ifr 1 = 1 r(t)is the position vector of a point for each value of tin some interval then,
given a cartesian coordinate system,
r(t) 1 = 1 x(t)i 1 + 1 y(t)j 1 + 1 z(t)k (16.24)
is a parametric representation of a curve C in
three-dimensional space, and tis the parameter of the
representation (Figure 16.17). The sense of increasing
values of tis called the positive sense on C, and defines a
direction on the curve.
When the parameter tis the time variable, derivatives
of position vectors provide a general method for the
description of the mechanics of dynamic systems.
EXAMPLE 16.9Velocity, acceleration, momentum and force
Velocity and acceleration
Ifr 1 = 1 r(t)is the position vector of a body in space, and the parameter tis the time
variable, thendr 2 dtis the velocity of the body,
v== (16.25)
+
+
=
d
dt
dx
dt
dy
dt
dz
dt
r
ijkvvv v
xy zijk++
d
dt
tte
d
dt
te
ta
ijk
a
=+ − =−ij+
−62 62
22cos , sin
−−tk
d
dt
da
dt
da
dt
da
dt
xyza
= ij
k
...
....
...
....
...
....
...
....
...
....
....
...
...
....
..C
o
z
y
x
r(t)
t
.
................................
...
......
.......
......
......
..
..
.............................................
............
..........
........
........
.......
......
.......
......
......
......
......
.....
......
......
......
.....
.......
.....
......
.......
......
......
.......
....
.....
.....
.....
.....
......
......
......
........
........
..........
...............
.........................................
...............
.........
........
.......
.....
......
.....
....
.....
.
..
........
..........
.........
........
..........
.........
........
..........
.........
.........
.........
.........
.........
.........
.........
.........
..........
........
.........
..........
........
.........
..........
........
.........
..........
........
.........
.....
......
......
......
......
......
.
.......
.......................
...
Figure 16.17