Biomedical Engineering Reference
In-Depth Information
a
e
Figure 1.1
The vector
a
=
ae
with
a >
0.
a
c
b
Figure 1.2
Graphical representation of the sum of two vectors:
c
=
a
+
b
.
a
and
b
is a new vector
The
sum
of two vectors
c
, equal to the diagonal of the
a
and
b
, see Fig.
1.2
:
parallelogram spanned by
c
=
a
+
b
.
(1.3)
This may be interpreted as follows. Imagine two thin wires which are attached
to a point P. The wires are being pulled at in two different directions according
to the vectors
a
and
b
. The length of each vector represents the magnitude of the
pulling force. The net force vector exerted on the attachment point P is the vector
sum of the two vectors
a
and
b
. If the wires are aligned with each other and the
pulling direction is the same, the resulting force direction is clearly coinciding
with the direction of the two wires and the length of the resulting force vector is
the sum of the two pulling forces. Alternatively, if the two wires are aligned but
the pulling forces are in opposite directions and of equal magnitude, the resulting
force exerted on point P is the zero vector 0.
The
inner product
or
dot product
of two vectors is a scalar quantity, defined
as
·
b
||
b
a
=|
a
|
cos(
φ
) ,
(1.4)
where
φ
is the smallest angle between
a
and
b
, see Fig.
1.3
. The inner product is
commutative
,i.e.
·
b
=
b
a
·
a
.
(1.5)
