Graphics Reference
In-Depth Information
Fig. 3.1
A vector is
represented by an oriented
line segment
3.3 Graphical Representation of Vectors
Cartesian coordinates provide an excellent mechanism for representing vectors and
allows them to be incorporated within the classical framework of mathematics. Fig-
ure
3.1
shows an oriented line segment used to represent a vector. The length of the
line represents the vector's magnitude, and the line's orientation and arrow define
its direction.
The line's direction is determined by the vector's head
(x
h
,y
h
)
and tail
(x
t
,y
t
)
from which we compute its
x
- and
y
-components
x
and
y
:
x
=
x
h
−
x
t
y
=
y
h
−
y
t
.
For example, in Fig.
3.1
the vector's head is
(
6
,
4
)
and its tail is
(
1
,
1
)
, which
makes its components
x
=
T
. If the vector is pointing in the
5 and
y
=
3or
[
53
]
T
.
One can readily see from this notation that a vector does not have a unique posi-
tion in space. It does not matter where we place a vector, so long as we preserve its
length and orientation its components will not alter.
opposite direction, its components become
x
=−
5 and
y
=−
3or
[−
5
−
3
]
3.4 Magnitude of a Vector
The length or
magnitude
of a vector
r
is written
and is computed by applying the
theorem of Pythagoras to its components
x
and
y
:
|
r
|
x
+
y
.
|
r
|=
is
√
3
2
T
For example, the magnitude of vector
5. Figure
3.2
shows
eight vectors, with their geometric properties listed in Table
3.1
. The subscripts
h
and
t
stand for
head
and
tail
respectively.
[
34
]
+
4
2
=