Graphics Reference
In-Depth Information
2.7 Rectangular unit vectors
A very powerful feature of vector algebra emerges from the previous definition of a vector, in that it
is possible to express a vector as the sumof unit vectors aligned with the rectangular Cartesian axes.
The following description moves to three dimensions but is equally applicable in two dimensions.
Note also that throughout this text we will employ a right-handed system of 3D axes.
Y
j
k
i
Z
X
Figure 2.8.
We begin by defining three
rectangular unit vectors
i
,
j
, and
k
, parallel with the x-, y-, and z-
rectangular Cartesian axes, respectively, as shown in Fig. 2.8, where
⎡
⎤
⎡
⎤
⎡
⎤
1
0
0
0
1
0
0
0
1
⎣
⎦
⎣
⎦
⎣
⎦
=
=
=
i
j
k
=
abc
T
Consequently, any vector
v
can be written as
v
=
a
i
+
b
j
+
c
k
which provides us with a simple algebraic mechanism for manipulating vectors.
For example, we can reverse
v
:
−
v
=−
a
i
−
b
j
−
c
k
We can double the length of
v
:
2
v
=
2a
i
+
2b
j
+
2c
k
And if we have two vectors,
v
=
2
i
+
3
j
+
4
k
and
w
=
5
i
+
6
j
+
7
k
we can add them together:
v
+
w
=
2
i
+
3
j
+
4
k
+
5
i
+
6
j
+
7
k
=
7
i
+
9
j
+
11
k
Soon we will discover how to multiply two vectors using this notation.
