Study sheet — Complex Numbers
Harry Dankowicz
Mechanical Science and Engineering
University of Illinois at UrbanaChampaign
[email protected]
Objective:
To investigate the relationship between transformations of the plane and complex numbers.
Observation 1
Consider a transformation
T
that rotates objects in the plane by
30
◦
and scales their size
by a factor of
5
3
. Suppose, for example, that the vector
v
makes an angle
45
◦
relative to the positive
horizontal axis of some coordinate system and has length
2
. Then, the result of applying the transfor
mation
T
to the vector
v
is the vector
T
(
v
)
that makes an angle
45
◦
+ 30
◦
= 75
◦
relative to the same
axis and has length
2
·
5
3
=
10
3
.
The matrix of projections of the vector
v
onto the coordinate axis equals
μ
2 cos 45
◦
2 sin 45
◦
¶
.
(1)
Similarly, the matrix of projections of the vector
T
(
v
)
onto the coordinate axes equals
μ
10
3
cos 75
◦
10
3
sin 75
◦
¶
.
(2)
But,
μ
10
3
cos 75
◦
10
3
sin 75
◦
¶
=
10
3
μ
cos (30
◦
+ 45
◦
)
sin (30
◦
+ 45
◦
)
¶
=
μ
5
3
·
2
¶ μ
cos 30
◦
cos 45
◦
−
sin 30
◦
sin 45
◦
sin 30
◦
cos 45
◦
+ cos 30
◦
sin 45
◦
¶
=
μ
5
3
cos 30
◦
−
5
3
sin 30
◦
5
3
sin 30
◦
5
3
cos 30
◦
¶ μ
2 cos 45
◦
2 sin 45
◦
¶
,
(3)
i.e., the matrix of projections of
T
(
v
)
onto the coordinate axes is obtained by premultiplying the
matrix of projections of
v
onto the coordinate axes with the matrix
μ
5
3
cos 30
◦
−
5
3
sin 30
◦
5
3
sin 30
◦
5
3
cos 30
◦
¶
.
(4)
HHHHH
Observation 2
Consider a transformation
T
that rotates objects in the plane by
30
◦
and scales their size
by a factor of
5
3
. Suppose, for example, that the vectors
v
and
w
make angles
30
◦
and
−
30
◦
, respec
tively, relative to the positive horizontal axis of some coordinate system and have lengths
1
2
and
3
2
,
1
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respectively. Then, the results of applying the transformation
T
to the vectors
v
and
w
are the vectors
T
(
v
)
and
T
(
w
)
that make angles
60
◦
and
0
◦
, respectively, relative to the same axis and have lengths
5
6
and
5
2
, respectively.
The vector sum
v
+
w
makes an angle
≈ −
16
.
10
◦
relative to the positive horizontal axis of the co
ordinate system and has length
√
13
2
.
The vector sum
T
(
v
) +
T
(
w
)
makes an angle
≈
13
.
90
◦
=
−
16
.
10
◦
+ 30
◦
relative to the same axis and has length
5
√
13
6
=
√
13
2
·
5
3
. It follows that, in this case,
T
(
v
) +
T
(
w
) =
T
(
v
+
w
)
.
(5)
Theory:
Denote by
T
(
r,
θ
)
a transformation that rotates objects in the plane by an angle
θ
and scales their
size by a factor
r
. Then,
T
(
r,
θ
)
is a
linear transformation
, i.e.,
T
(
r,
θ
)
(
α
v
+
β
w
) =
α
T
(
r,
θ
)
(
v
) +
β
T
(
r,
θ
)
(
w
)
,
(6)
where
v
and
w
are arbitrary vectors and
α
and
β
are arbitrary real numbers.
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 Spring '08
 Weaver
 Cartesian Coordinate System, Sin, Cos, Complex number

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