Biomedical Engineering Reference
In-Depth Information
1.4
Decomposition of a vector with respect to a basis
As stated above, a Cartesian vector basis is an orthonormal basis. Any vector can
be decomposed into the sum of, at most, three vectors parallel to the three basis
vectors
e
x
,
e
y
and
e
z
:
a
=
a
x
e
x
+
a
y
e
y
+
a
z
e
z
.
(1.19)
The components
a
x
,
a
y
and
a
z
can be found by taking the inner product of the
vector
a
with respect to each of the basis vectors:
a
x
=
a
·
e
x
a
y
=
a
·
e
y
(1.20)
a
z
=
a
·
e
z
,
where use is made of the fact that the basis vectors have unit length and are
mutually orthogonal, for example:
a
·
e
x
=
a
x
e
x
·
e
x
+
a
y
e
y
·
e
x
+
a
z
e
z
·
e
x
=
a
x
.
(1.21)
a
with respect to the Cartesian
vector basis, may be collected in a
column
, denoted by
∼
:
The components, say
a
x
,
a
y
and
a
z
, of a vector
⎡
⎤
a
x
a
y
a
z
⎣
⎦
∼
=
.
(1.22)
So, with respect to a Cartesian vector basis any vector
a
may be decomposed in
components that can be collected in a column:
a
←→
∼
.
(1.23)
This 'transformation' is only possible and meaningful if the vector basis with
which the components of the column
∼
are defined has been specified. The choice
of a different vector basis leads to a different column representation
∼
of the vector
a
, this is illustrated in Fig.
1.4
. The vector
a
has two different column representa-
tions,
∼
and
∼
∗
, depending on which vector basis is used. If, in a two-dimensional
context
{
e
x
,
e
y
}
is used as a vector basis then
a
x
a
y
,
a
−→
∼
=
(1.24)
e
x
,
e
y
}
while, if
{
is used as vector basis:
a
x
a
y
.
∼
∗
=
a
−→
(1.25)
