Biomedical Engineering Reference
In-Depth Information
Clearly, if two force vectors
F
1
and
F
2
are parallel, then the resulting force
vector
F
3
=
F
1
+
F
2
will be parallel to the vectors
F
1
and
F
2
as well. If
F
1
=−
F
2
,
then the addition of these two force vectors yields the so-called zero vector 0,
having zero length.
2.5
Force decomposition
Suppose that a bone is loaded with a force
F
as sketched in Fig.
2.4
. The principal
axis of the bone has a direction indicated by the unit vector
e
. The smallest angle
between the force vector
F
and the unit vector
e
is denoted by
α
. It is useful to
know, which part of the force
F
acts in the direction of the unit vector
e
, indicated
by
F
t
and which part of the force acts perpendicular to the bone, indicated by the
force vector
F
n
. The force vector
F
may, in that case, be written as
F
=
F
t
+
F
n
.
(2.13)
To determine the vectors
F
t
and
F
n
, vector calculus will be used. The inner product
of two vectors, say
a
and
b
, is defined as
·
b
||
b
a
=|
a
|
cos(
α
) ,
(2.14)
where
α
is the smallest angle between the two vectors
a
and
b
, see Fig.
2.5
and
Chapter
1
for further details on the properties of the vector inner product. Com-
putation of the inner product requires knowledge of the length of both vectors
F
F
n
α
e
F
t
Figure 2.4
Bone loaded by the force vector
F
. The orientation of the bone is indicated by the unit vector
e
.
b
α
a
Figure 2.5
Definition of the angle
α
.
