Biomedical Engineering Reference
In-Depth Information
e
y
*
e
y
a
a
a
y
a
y
*
e
x
*
a
x
*
a
x
e
x
Figure 1.4
Vector
a
with respect to vector basis {
e
x
,
e
y
} and {
e
x
,
e
y
}.
Consequently, with respect to a Cartesian vector basis, vector operations such as
multiplication, addition, inner product and dyadic product may be rewritten as
'column' (actually matrix) operations.
Multiplication of a vector
a
=
a
x
e
x
+
a
y
e
y
+
a
z
e
z
with a scalar
α
yields a new
vector, say
b
:
b
=
α
a
=
α
(
a
x
e
x
+
a
y
e
y
+
a
z
e
z
)
=
α
a
x
e
x
+
α
a
y
e
y
+
α
a
z
e
z
.
(1.26)
So
b
=
α
a
−→
∼
=
α
∼
.
(1.27)
The sum of two vectors
a
and
b
leads to
c
=
a
+
b
−→
∼
=
∼
+
∼
.
(1.28)
Using the fact that the Cartesian basis vectors have unit length and are mutually
orthogonal, the inner product of two vectors
a
and
b
yields a scalar
c
according to
c
=
a
·
b
=
(
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
x
b
x
+
a
y
b
y
+
a
z
b
z
.
(1.29)
In column notation this result is obtained via
T
∼
,
c
=
∼
(1.30)
T
denotes the
transpose
of the column
∼
, defined as
where
∼
=
a
x
a
y
a
z
,
T
∼
(1.31)
such that:
⎡
⎣
⎤
⎦
=
b
x
b
y
b
z
T
∼
=
a
x
a
y
a
z
∼
a
x
b
x
+
a
y
b
y
+
a
z
b
z
.
(1.32)
