Biomedical Engineering Reference
In-Depth Information
b
φ
a
Figure 1.3
Definition of the angle
φ
.
The inner product can be used to define the length of a vector, since the inner
product of a vector with itself yields (
φ
=
0):
2
.
a
·
a
=|
a
||
a
|
cos( 0)
=|
a
|
(1.6)
If two vectors are perpendicular to each other the inner product of these two
vectors is equal to zero, since in that case
φ
=
2
:
a
·
b
=
0, if
φ
=
2
.
(1.7)
a
and
b
yields a new vector
The
cross product
or
vector product
of two vectors
a
and
b
such that
a
,
b
and
c
that is perpendicular to both
c
form a right-handed
system. The vector
c
is denoted as
×
b
.
c
=
a
(1.8)
The length of the vector
c
is given by
|
c
|=|
a
||
b
|
sin(
φ
) ,
(1.9)
a
and
b
. The length of
where
φ
is the smallest angle between
c
equals the area of
a
and
b
. The vector system
a
,
b
and
the parallelogram spanned by the vectors
c
forms a right-handed system, meaning that if a corkscrew is used rotating from
a
to
b
the corkscrew would move into the direction of
c
.
The vector product of a vector
a
with itself yields the zero vector since in that
case
φ
=
0:
=
0.
a
×
a
(1.10)
The vector product is
not
commutative, since the vector product of
b
and
a
yields
a
and
b
:
a vector that has the opposite direction of the vector product of
a
×
b
=−
b
×
a
.
(1.11)
The
triple product
of three vectors
a
,
b
and
c
is a scalar, defined by
×
b
×
b
)
a
·
c
=
(
a
·
c
.
(1.12)
a
and
b
is determined and subsequently the inner
product of the resulting vector with the third vector
So, first the vector product of
c
is taken. If all three vectors
a
,
b
and
c
are non-zero vectors, while the triple product is equal to zero then the
