Biomedical Engineering Reference
In-Depth Information
is acute, the vector
b
t
has the same sense as the unit vector
Since the angle
α
e
such that:
b
t
=|
b
t
|
e
=
(
b
·
e
)
e
.
(2.20)
If the angle
α
is obtuse, see Fig.
2.6
(b), hence if
α>π/
2, we have
=
|
b
t
|
|
b
cos(
π
−
α
)
=−
cos(
α
)
.
(2.21)
|
b
·
e
|
b
|
With, according to Eq. (
2.15
), cos(
α
)
=
this leads to
|
b
t
|=−
b
·
e
.
(2.22)
In this case the sense of the vector
b
t
is opposite to the unit vector
e
, such that
b
t
=−|
b
t
|
e
.
(2.23)
So, clearly, whether the angle
α
is acute or obtuse, the vector
b
t
parallel to the unit
vector
e
is given by
b
t
=
(
b
·
e
)
e
.
(2.24)
Recall, that this is only true if
e
has unit length! In conclusion, the inner product
of an arbitrary vector
b
with a unit vector
e
defines the magnitude and sense of a
vector
b
t
that is parallel to the unit vector
e
such that the original vector
b
may be
written as the sum of this parallel vector and a vector normal to the unit vector
e
.
The vector
b
n
normal to
e
follows automatically from
b
n
=
b
−
b
t
.
(2.25)
This implicitly defines the unique decomposition of the vector
b
into a component
normal and a component parallel to the unit vector
e
.
Based on the considerations above, the force vector
F
in Fig.
2.4
can be
decomposed into a component parallel to the bone principal axis
F
t
given by
F
t
=
(
F
·
e
)
e
,
(2.26)
where
e
denotes a vector of unit length, and a component normal to the principal
axis of the bone:
F
n
=
F
−
F
t
.
(2.27)
