Biomedical Engineering Reference
In-Depth Information
y
F
F
y
θ
F
x
x
FIGURE 4.2
Two-dimensional representation of vector
F
.
4.2.1 Vector Mathematics
Forces may be written in terms of scalar components and unit vectors, of magnitude
equal to one, or in polar form with magnitude and direction. Figure 4.2 shows that the
two-dimensional vector
F
is composed of the
i
component,
F
x
, in the
x
-direction, and
the
j
component,
F
y
, in the
y
-direction, or
F
¼
F
x
i
þ
F
y
j
ð
4
:
1
Þ
as in 20
i
40
j
lb. In this chapter, vectors are set in bold type. This same vector may be
written in polar form in terms of the vector's magnitude
þ
j
F
j
, also called the
norm
, and the
vector's angle of orientation, y:
q
F
j
F
j¼
2
x
þ
F
2
y
ð
4
:
2
Þ
arctan
F
y
F
x
y
¼
ð
4
:
3
Þ
4
. Vectors are similarly represented in three dimensions in
terms of their
i
,
j
, and
k
components:
yielding
j
F
j¼
44
:
7 lb and y
¼
63
:
F
¼
F
x
i
þ
F
y
j
þ
F
z
k
ð
4
:
4
Þ
with
k
in the
-direction.
Often, a vector's magnitude and two points along its line of action are known. Consider
the three-dimensional vector in Figure 4.3.
F
has magnitude of 10 lb, and its line of action
passes from the origin (0,0,0) to the point (2,6,4).
F
is written as the product of the magni-
tude
z
j
F
j
and a unit vector
e
F
that points along its line of action:
F
¼j
F
j
e
F
0
@
1
A
2
i
4
k
2
2
þ
6
j
þ
¼
10 lb
p
6
2
4
2
þ
þ
F
¼
2
:
67
i
þ
8
:
02
j
þ
5
:
34
k
lb
