Biomedical Engineering Reference
In-Depth Information
F
t
=
(
F
·
e
)
e
14
√
13
1
√
13
(2
=
−
e
x
−
3
e
y
)
F
·
e
e
14
13
(2
e
x
−
3
e
y
).
=−
2.7
Column notation
An arbitrary two-dimensional vector is written as
F
=
F
x
e
x
+
F
y
e
y
,
(2.38)
and in a three-dimensional space:
F
=
F
x
e
x
+
F
y
e
y
+
F
z
e
z
.
(2.39)
The numbers
F
x
,
F
y
and, in three dimensions
F
x
,
F
y
and
F
z
may be collected in a
column
∼
:
⎡
⎣
⎤
⎦
F
x
F
y
,
F
x
F
y
F
z
∼
=
∼
=
.
(2.40)
These numbers only have a meaning when associated with a vector basis, in this
case the Cartesian vector basis. There is a distinct difference between the vector
F
and the column
∼
. The vector
F
is independent of the choice of the vector basis,
while the numbers that are stored in the column
∼
depend on the vector basis that
has been chosen. An example is given in Fig.
2.10
. Both the vector basis
{
e
x
,
e
y
}
and the vector basis
{
e
x
,
e
y
}
are Cartesian, but they have a different orientation
in space. Hence, the column associated with each of these bases is different. With
respect to the
{
e
x
,
e
y
}
basis it holds that
F
x
F
y
,
∼
=
(2.41)
e
x
e
y
}
while with respect to
{
,
this column is given by
F
x
F
y
.
∼
∗
=
(2.42)
{
e
x
e
y
}
According to Fig.
2.10
the
,
basis is rotated by an angle
α
with respect to
the
{
e
x
,
e
y
}
basis. In that case:
