Biomedical Engineering Reference
In-Depth Information
·
b
=
b
since
a
. This demonstrates that if vectors are expressed with respect to
a vector basis, typical vector operations (such as vector addition and vector inner
product) are relatively straightforward to perform. If the vector sum of the basis
vectors and the inner products of the basis vectors with respect to each other and
themselves are known, vector operations on other vectors are straightforward.
However, expressing an arbitrary vector with respect to the basis vectors may
be cumbersome. For a given vector
F
this requires to determine the coefficients
a
·
α
and
β
in
F
+
β
b
.
=
α
a
(2.34)
One possibility to realize this, is to take the inner product of
F
with respect to both
the basis vectors:
F
·
a
=
α
a
·
a
+
β
b
·
a
F
·
b
=
α
a
·
b
+
β
b
·
b
.
(2.35)
Recall, that each of the above inner products can be computed and simply yield
a number. Therefore, the set of Eqs. (
2.35
) provides two linear equations from
which the two unknown coefficients
α
β
can be obtained. A similar operation
(involving three basis vectors) is needed in a three-dimensional space.
Solving for the coefficients
and
α
and
β
would be easy if the basis vectors
a
and
b
have unit length (such that e.g.
a
·
a
=
1) and if the basis vectors are mutu-
·
b
ally perpendicular, hence if
a
=
0. In that case, the set of Eqs. (
2.35
) would
reduce to:
F
·
a
=
α
F
·
b
=
β
.
(2.36)
If the vectors of a vector basis are mutually perpendicular but do not have unit
length, the vector basis is called orthogonal. If the vectors of an orthogonal vector
basis have unit length, then it is called an orthonormal vector basis. If the basis
vectors of an orthonormal basis have a so-called right-handed orientation with
respect to each other and are independent of the location in three-dimensional
space, it is called a Cartesian vector basis.
The
Cartesian
vector basis
{
e
x
,
e
y
,
e
z
}
is used to uniquely specify an arbitrary
vector, see Fig.
2.8
. An arbitrary force vector in two-dimensional space, say
F
,
can be expressed with respect to the Cartesian vector basis as
F
=
F
x
e
x
+
F
y
e
y
.
(2.37)
The use of a Cartesian vector basis substantially simplifies vector manipulation as
is illustrated next.
