Biomedical Engineering Reference
In-Depth Information
z
F
θ
z
4ft
θ
y
y
θ
x
2ft
6ft
x
FIGURE 4.3
Three-dimensional vector defined by its magnitude and line of action.
The quantity in parentheses is the unit vector of
F
,or
!
2
i
4
k
2
2
þ
6
j
þ
p
e
F
¼
¼
0
:
267
i
þ
0
:
802
j
þ
0
:
534
k
6
2
4
2
þ
þ
and the magnitude of
F
is
p
2
67
2
02
2
34
2
j
F
j¼
:
þ
8
:
þ
5
:
¼
10 lb
The vector
F
in Figure 4.3 may also be defined in 3-D space in terms of the angles between
its line of action and each coordinate axis. Consider the angles y
x
, y
y
, and y
z
that are
measured from the positive
x
,
y
, and
z
axes, respectively, to
F
. Then
cos y
x
¼
F
x
j
F
j
ð
4
:
5
Þ
cos y
y
¼
F
y
j
F
j
ð
4
:
6
Þ
cos y
z
¼
F
z
j
F
j
ð
4
:
7
Þ
These ratios are termed the
direction cosines
of
F
. The unit vector
e
F
is equivalent to
e
F
¼
cos y
x
i
þ
cos y
y
j
þ
cos y
z
k
ð
4
:
8
Þ
or, in general
0
@
1
A
F
x
i
k
F
þ
F
y
j
þ
F
z
e
F
¼
q
ð
4
:
9
Þ
2
2
2
z
x
þ
F
y
þ
F