Biomedical Engineering Reference
In-Depth Information
|
b
(i.e.
|
a
|
and
|
) and the smallest angle between the two vectors (i.e.
α
), all physi-
a
and
b
are perpendicular
cal quantities that can easily be obtained. If the vectors
·
b
to each other, hence if
α
=
π/
2, then th
e inn
er product equals zero, i.e.
a
=
0.
|=
√
The length of a vector satisfies
a
.
Now, consider the inner product of an arbitrary vector
b
with a unit vector
|
a
a
·
e
(i.e.
|
e
|=
1), then
b
·
e
=|
b
|
cos(
α
) .
(2.15)
Let the vector
b
be written as the sum of a vector parallel to
e
, say
b
t
, and a vector
normal to
e
, say
b
n
, such that:
b
=
b
t
+
b
n
,
(2.16)
as depicted in Fig.
2.6
(a).
If the angle
b
is acute, hence if
α
between the unit vector
e
and the vector
α
≤
π/
2, it is easy to show that this inner product is equal to the length of the
b
t
, the component of
b
parallel to the unit vector
vector
e
, see Fig.
2.6
(a). By
definition
cos(
α
)
=
|
b
t
|
|
b
|
.
(2.17)
However, from Eq. (
2.15
) we know that
b
·
e
|
b
|
cos(
α
)
=
,
(2.18)
hence
|
b
t
|=
b
·
e
.
(2.19)
b
n
b
b
b
n
α
α
e
e
b
t
b
t
|
b
t
|
|
b
t
|
α
, the length of
b
t
α
, the length of
b
t
(a) Acute angle
(b) Obtuse angle
Figure 2.6
Vector decomposition in case of (a) an acute and (b) an obtuse angle between the vectors.
