Graphics Reference
In-Depth Information
x
2
+
y
2
+
z
2
. For example, the point
P(
4
,
5
,
6
)
creates a position vector
p
relative
to the origin:
4
2
T
5
2
6
2
p
=[
456
]
and
|
p
|=
+
+
≈
20
.
88
.
3.8 Unit Vectors
By definition, a
unit vector
has a length of 1. A simple example is
i
where
T
i
=[
100
]
and
|
i
|=
1
.
Converting a vector into a unit form is called
normalising
and is achieved by di-
viding the vector's components by its lengt
h. To formali
se this process consider the
vector
r
x
2
T
with length
y
2
z
2
. The unit form of
r
is given
=[
xyz
]
|
r
|=
+
+
by
1
T
.
ˆ
r
=
|
[
xyz
]
|
r
ˆ
This process is confirmed by showing that the length of
r
is 1:
x
|ˆ
2
y
|ˆ
2
z
|ˆ
2
|ˆ
r
|=
+
+
r
|
r
|
r
|
x
2
1
|ˆ
=
+
y
2
+
z
2
|
r
|ˆ
r
|=
1
.
T
into a unit form:
To put this into context, consider the conversion of
r
=[
123
]
1
2
√
14
2
2
3
2
|
r
|=
+
+
=
⎡
⎤
⎡
⎤
1
2
3
0
.
267
0
.
535
0
.
802
1
√
14
⎣
⎦
≈
⎣
⎦
.
r
ˆ
=
3.9 Cartesian Vectors
We begin by defining three Cartesian unit vectors
i
,
j
,
k
aligned with the
x
-,
y
- and
z
-axes respectively:
⎡
⎤
⎡
⎤
⎡
⎤
1
0
0
0
1
0
0
0
1
⎣
⎦
,
⎣
⎦
,
⎣
⎦
.
i
=
j
=
k
=
Any vector aligned with the
x
-,
y
-or
z
-axes can be defined by a scalar multiple of
the unit vectors
i
,
j
and
k
respectively. For example, a vector 10 units long aligned
