Biomedical Engineering Reference
In-Depth Information
Using the properties of the basis vectors of the Cartesian vector basis:
e
x
×
e
x
=
0
e
x
×
e
y
=
e
z
e
x
×
e
z
=−
e
y
e
y
×
e
x
=−
e
z
e
y
×
e
y
=
0
(1.33)
e
y
×
e
z
=
e
x
e
z
×
e
x
=
e
y
e
z
×
e
y
=−
e
x
e
z
=
0,
the vector product of a vector
a
and a vector
b
is directly computed by means of
e
z
×
×
b
a
=
(
a
x
e
x
+
a
y
e
y
+
a
z
e
z
)
×
(
b
x
e
x
+
b
y
e
y
+
b
z
e
z
)
=
(
a
y
b
z
−
a
z
b
y
)
e
x
+
(
a
z
b
x
−
a
x
b
z
)
e
y
+
(
a
x
b
y
−
a
y
b
x
)
e
z
.
(1.34)
If by definition
c
=
a
×
b
, then the associated column
∼
can be written as:
⎡
⎣
⎤
⎦
a
y
b
z
−
a
z
b
y
∼
=
a
z
b
x
−
a
x
b
z
.
(1.35)
a
x
b
y
−
a
y
b
x
The dyadic product
ab
transforms another vector
c
into a vector
d
, according to
the definition
d
ab
=
·
c
=
A
·
c
,
(1.36)
with
A
the second-order tensor equal to the dyadic product
ab
. In column notation
this is equivalent to
T
∼
)
T
)
∼
,
∼
=
∼
(
∼
=
(
∼ ∼
(1.37)
T
a(3
×
3) matrix given by
with
∼ ∼
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
a
x
a
y
a
z
a
x
b
x
a
x
b
y
a
x
b
z
b
x
b
z
T
A
=
∼ ∼
=
b
y
=
a
y
b
x
a
y
b
y
a
y
b
z
,
(1.38)
a
z
b
x
a
z
b
y
a
z
b
z
or
∼
=
A
∼
.
(1.39)
