Biomedical Engineering Reference
In-Depth Information
2.6
Representation of a vector with respect to a vector basis
Two vectors are called mutually independent if both vectors are non-zero and
non-parallel. In a two-dimensional space any vector can be expressed as a linear
combination of two mutually independent vectors, say
a
and
b
, by using the scalar
product and vector sum, for example:
F
=
α
a
+
β
b
,
(2.28)
see, Fig.
2.7
. Clearly, in a three-dimensional space three mutually independent
vectors are needed:
F
+
β
b
c
. (2.29)
The above mutually independent vectors
a
and
b
are called basis vectors that
form a so-called vector basis
{
a
,
b
}
in a two-dimensional space, while in a three-
dimensional space three mutually independent vectors, say
a
,
b
and
c
, are needed
to form a basis. It is convenient to have such a vector basis because it facilitates
vector manipulation. For example, let
F
1
=
=
α
a
+
γ
5
b
,
2
a
+
(2.30)
and
F
2
=−
a
+
3
b
,
(2.31)
then
F
1
+
F
2
=
a
+
8
b
,
F
1
+
3
F
2
=−
a
+
14
b
.
(2.32)
The inner product of
F
1
and
F
2
yields
F
1
·
F
2
=
(2
a
+
5
b
)
·
(
−
a
+
3
b
)
=−
a
·
b
−
5
b
·
a
+
15
b
·
b
a
·
a
+
2
6
a
·
a
+
a
·
b
+
15
b
·
b
,
=−
2
(2.33)
a
α
a
b
β
b
F
Figure 2.7
An arbitrary vector
F
as a linear combination of two vectors:
F
=
αa
+
β b
.
