Biomedical Engineering Reference
In-Depth Information
The angles y
x
, y
y
, and y
z
for this example are consequently
arccos
2
!
67
10
:
5
y
x
¼
¼
74
:
arccos
8
!
02
10
:
7
y
y
¼
¼
36
:
arccos
5
!
34
10
:
7
y
z
¼
¼
57
:
Vectors are added by summing their components:
A
¼
A
x
i
þ
A
y
j
þ
A
z
k
B
¼
B
x
i
þ
B
y
j
þ
B
z
k
C
¼
A
þ
B
¼ð
A
x
þ
B
x
Þ
i
þð
A
y
þ
B
y
Þ
j
þð
A
z
þ
B
z
Þ
k
In general, a set of forces may be combined into an equivalent force denoted the resultant
R
, where
X
X
X
R
¼
F
x
i
þ
F
y
j
þ
F
z
k
ð
4
:
10
Þ
as will be illustrated in subsequent sections. Vectors are subtracted similarly by subtracting
vector components.
Vector multiplication consists of two distinct operations: the
dot
and
cross
products.
The dot, or scalar, product of vectors
A
and
B
produces a scalar via
A
B
¼
AB
cos y
ð
4
:
11
Þ
where y is the angle between the vectors. For an orthogonal coordinate system, where all
axes are 90
apart, all like terms alone remain, since
i
i
¼
j
j
¼
k
k
¼
1
ð
4
:
12
Þ
i
j
¼
j
k
¼
k
i
¼¼
0
For example:
A
¼
3
i
þ
2
j
þ
k
ft
B
¼
2
i
þ
3
j
þ
10
k
lb
A
B
¼
3
ð
2
Þþ
2
ð
3
Þþ
1
ð
10
Þ¼
10 ft lb
Note that the dot product is commutative—that is,
A
A
.
B
B
The physical interpretation of the dot product
A
B
is the projection of
A
onto
B
, or,
equivalently, the projection of
B
onto
A
. For example,
is defined as the force that acts
in the same direction as the motion of a body. Figure 4.4 (left) shows a force vector
F
dotted
with a direction of motion vector
d
. The work
work
W
done by
F
is given by
F
d
Fd
cos y.
Dotting
F
with
d
yields the component of
F
acting in the same direction as
d
.
