Graphics Reference
In-Depth Information
Z'
Y
(2, 2, 0)
Y'
(1, 1, 0)
X
'
(1, 0, 1)'
Z
X
Fig. 7.29.
Vector [1 1 0]
T
is transformed to [
−
10
−
1]
T
.
linear equations such as
c
1
=
a
1
x
+
b
1
y
c
2
=
a
2
x
+
b
2
y
(7.106)
where values of
x
and
y
are defined in terms of the other constants. Without
showing the solution, the values of
x
and
y
are given by
c
1
b
2
−
c
2
b
1
x
=
a
1
b
2
−
a
2
b
1
y
=
a
1
c
2
−
a
2
c
1
(7.107)
a
1
b
2
−
a
2
b
1
provided that the denominator
a
1
b
2
−
=0.
It is also possible to write the linear equations in matrix form as
c
1
c
2
a
2
b
1
=
a
1
x
y
b
1
·
(7.108)
a
2
b
2
and we notice that the denominator comes from the matrix terms
a
1
b
2
−
a
2
b
1
.Thisiscalledthe
determinant
, and is valid only for square matrices.
A determinant is defined as follows:
a
1
b
1
=
a
1
b
2
−
a
2
b
1
(7.109)
a
2
b
2
With this notation it is possible to rewrite the original linear equations as
x
y
1
=
=
(7.110)
c
1
b
1
a
1
c
1
a
1
b
1
c
2
b
2
a
2
c
2
a
2
b
2
With a set of three linear equations:
d
1
=
a
1
x
+
b
1
y
+
c
1
z
d
2
=
a
2
x
+
b
2
y
+
c
2
z
d
3
=
a
3
x
+
b
3
y
+
c
3
z
(7.111)
