Biomedical Engineering Reference
In-Depth Information
*
e
y
*
e
y
e
y
e
y
F
F
y
F
e
x
*
*
e
x
*
F
y
α
*
F
x
e
x
e
x
F
x
(a)
(b)
(c)
Figure 2.10
(a) Basis rotation by angle
α
(b)
F
with respect to basis {
e
x
,
e
y
} (c)
F
with respect to basis {
e
x
,
e
y
}.
e
x
e
y
e
x
=
α
−
α
cos(
)
sin(
)
e
x
e
y
e
y
=
sin(
α
)
+
cos(
α
)
,
(2.43)
or, alternatively:
e
x
=+
cos(
α
)
e
x
+
sin(
α
)
e
y
e
y
=−
sin(
α
)
e
x
+
cos(
α
)
e
y
.
(2.44)
Therefore, if
F
is known with respect to the
{
e
x
,
e
y
}
basis, i.e.
F
=
F
x
e
x
+
F
y
e
y
,
(2.45)
e
x
,
e
y
}
this vector can also be expressed with respect to the
{
basis:
F
x
cos(
e
y
F
y
sin(
e
y
F
e
x
e
x
=
α
)
−
sin(
α
)
+
α
)
+
cos(
α
)
=
F
x
cos(
α
)
+
F
y
sin(
α
)
e
x
+
−
F
x
sin(
α
)
+
F
y
cos(
α
)
e
y
.
(2.46)
Therefore, in terms of
F
x
and
F
y
:
F
x
cos(
.
α
+
F
y
sin(
α
)
)
∼
∗
=
(2.47)
−
F
x
sin(
α
)
+
F
y
cos(
α
)
|
F
|
The magnitude
of the force vector is obtained from
F
·
F
.
|
F
|=
(2.48)
In a three-dimensional context, it follows immediately that with respect to a
Cartesian vector basis:
F
x
+
F
y
+
F
z
,
|
F
|=
(2.49)
or, equivalently
∼
|
F
|=
T
∼
.
(2.50)
