x2 t01 09 geometrical representation (2011)
TRANSCRIPT
Geometrical Representation of
Complex Numbers
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
iyxz
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
iyxz
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
iyxz
The advantage of using vectors is that they can be moved around the
Argand Diagram
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
iyxz
The advantage of using vectors is that they can be moved around the
Argand Diagram
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
iyxz
The advantage of using vectors is that they can be moved around the
Argand Diagram
No matter where the vector is placed its length (modulus) and the angle
made with the x axis (argument) is constant
Geometrical Representation of
Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
y
x
iyxz
The advantage of using vectors is that they can be moved around the
Argand Diagram
No matter where the vector is placed its length (modulus) and the angle
made with the x axis (argument) is constant
A vector always
represents
HEAD minus TAIL
Addition / Subtraction
Addition / Subtraction
y
x
Addition / Subtraction
y
x
1z
Addition / Subtraction
y
x
1z
2z
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz 2121 and
diagonals; twohas vectors
addingby formed ramparallelog the
:
zzzz
NOTE
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz 2121 and
diagonals; twohas vectors
addingby formed ramparallelog the
:
zzzz
NOTE
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz 2121 and
diagonals; twohas vectors
addingby formed ramparallelog the
:
zzzz
NOTE
Trianglar Inequality
In any triangle a side will be shorter than the sum of the other two sides
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz 2121 and
diagonals; twohas vectors
addingby formed ramparallelog the
:
zzzz
NOTE
Trianglar Inequality
In any triangle a side will be shorter than the sum of the other two sides
BCABACABC ;In
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz 2121 and
diagonals; twohas vectors
addingby formed ramparallelog the
:
zzzz
NOTE
Trianglar Inequality
In any triangle a side will be shorter than the sum of the other two sides
BCABACABC ;In (equality occurs when AC is a straight line)
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
To subtract two complex numbers, place the vectors “head to head” (or
add the negative vector)
21 zz 2121 and
diagonals; twohas vectors
addingby formed ramparallelog the
:
zzzz
NOTE
Trianglar Inequality
In any triangle a side will be shorter than the sum of the other two sides
BCABACABC ;In (equality occurs when AC is a straight line)
2121 zzzz
Addition
If a point A represents and point B
represents then point C representing
is such that the points OACB
form a parallelogram.
1z
2z
21 zz
Addition
If a point A represents and point B
represents then point C representing
is such that the points OACB
form a parallelogram.
1z
2z
21 zz
Subtraction
If a point D represents
and point E represents
then the points ODEB form a
parallelogram.
Note:
1z
12 zz
12 zzAB
12arg zz
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
wOQ is oflength The
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
wOQ is oflength The
wzPQ is oflength The
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
wOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangula theUsing
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
wOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangula theUsing
(ii) Construct the point R representing z + w, What can be said about the
quadrilateral OPRQ?
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
wOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangula theUsing
(ii) Construct the point R representing z + w, What can be said about the
quadrilateral OPRQ?
R
1995..ge y
x
P
O
Q
The diagram shows a complex plane with origin O.
The points P and Q represent the complex numbers z and w respectively.
Thus the length of PQ is wz
wzwzi that Show
zOP is oflength The
wOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangula theUsing
(ii) Construct the point R representing z + w, What can be said about the
quadrilateral OPRQ?
R
OPRQ is a parallelogram
?about said becan what , If z
wwzwziii
?about said becan what , If z
wwzwziii
wzwz
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
w
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
wimaginarypurely is
z
w
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
wimaginarypurely is
z
w
Multiplication
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
wimaginarypurely is
z
w
Multiplication
2121 zzzz
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
wimaginarypurely is
z
w
Multiplication
2121 zzzz 2121 argargarg zzzz
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
wimaginarypurely is
z
w
Multiplication
2121 zzzz 2121 argargarg zzzz
21212211 cisrrcisrcisr
?about said becan what , If z
wwzwziii
wzwz i.e. diagonals in OPRQ are =
rectangle a is OPRQ
2argarg
zw
2arg
z
wimaginarypurely is
z
w
Multiplication
2121 zzzz 2121 argargarg zzzz
21212211 cisrrcisrcisr
2
2121
by multiplied islength its and
by iseanticlockw rotated is vector the,by multiply weif ..
r
zzzei
If we multiply by the vector
OA will rotate by an angle of in an
anti-clockwise direction. If we
multiply by it will also multiply
the length of OA by a factor of r
1z cis
rcis
If we multiply by the vector
OA will rotate by an angle of in an
anti-clockwise direction. If we
multiply by it will also multiply
the length of OA by a factor of r
1z cis
rcis
ii 2
sin2
cos
Note: 1iz will rotate OA anticlockwise 90 degrees.
If we multiply by the vector
OA will rotate by an angle of in an
anti-clockwise direction. If we
multiply by it will also multiply
the length of OA by a factor of r
1z cis
rcis
ii 2
sin2
cos
Note: 1iz will rotate OA anticlockwise 90 degrees.
2by iseanticlockwrotation a is by tion Multiplica
i
If we multiply by the vector
OA will rotate by an angle of in an
anti-clockwise direction. If we
multiply by it will also multiply
the length of OA by a factor of r
1z cis
rcis
ii 2
sin2
cos
Note: 1iz will rotate OA anticlockwise 90 degrees.
REMEMBER: A vector is HEAD minus TAIL
2by iseanticlockwrotation a is by tion Multiplica
i
y
x
A
O
C
B
)1(D
y
x
A
O
C
B
)1(D
DC DA i
y
x
A
O
C
B
)1(D
DC DA i
1 1C i
y
x
A
O
C
B
)1(D
DC DA i
1 1C i
1 1
1
C i
i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1 1C i
1 1
1
C i
i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
1 1C i
1 1
1
C i
i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
1 1C i
1 1
1
C i
i i
1
1
B i
i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 1C i
1 1
1
C i
i i
1
1
B i
i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
OR
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
OR
24
DB cis DA
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
OR
24
DB cis DA
1 2 cos sin ( 1)4 4
B i
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
OR
24
DB cis DA
1 2 cos sin ( 1)4 4
B i
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
1 ( 1) 1B i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
OR
24
DB cis DA
1 2 cos sin ( 1)4 4
B i
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
1 ( 1) 1B i
1 1i i
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1B C
OR
1 ( 1)B i i
OR
24
DB cis DA
1 2 cos sin ( 1)4 4
B i
1 1C i
1 1
1
C i
i i
1
1
B i
i i
B C DA
1i i
1 ( 1) 1B i
1 1i i
1i i
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
2C i
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
(ii) What complex number corresponds to the point of intersection D of
the diagonals OB and AC?
2C i
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
(ii) What complex number corresponds to the point of intersection D of
the diagonals OB and AC?
diagonals bisect in a rectangle
2C i
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
(ii) What complex number corresponds to the point of intersection D of
the diagonals OB and AC?
diagonals bisect in a rectangle
2C i
midpoint of D AC
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
(ii) What complex number corresponds to the point of intersection D of
the diagonals OB and AC?
diagonals bisect in a rectangle
2C i
midpoint of D AC
2
A CD
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
(ii) What complex number corresponds to the point of intersection D of
the diagonals OB and AC?
diagonals bisect in a rectangle
2C i
midpoint of D AC
2
A CD
2
2
iD
2000..ge y
x
A
O
C
In the Argand Diagram, OABC is a rectangle, where OC = 2OA.
The vertex A corresponds to the complex number
B
? toscorrespondnumber complex What Ci
2OC OA i
(ii) What complex number corresponds to the point of intersection D of
the diagonals OB and AC?
diagonals bisect in a rectangle
1
2D i
2C i
midpoint of D AC
2
A CD
2
2
iD
Examples
Examples
3OB OA cis
Examples
3OB OA cis
13
B cis
Examples
3OB OA cis
iB2
3
2
1
13
B cis
Examples
3OB OA cis
iB2
3
2
1
13
B cis
OD OB i
Examples
3OB OA cis
iB2
3
2
1
iBD
13
B cis
OD OB i
Examples
3OB OA cis
iB2
3
2
1
iBD
iD2
1
2
3
13
B cis
OD OB i
Examples
3OB OA cis
iB2
3
2
1
iBD
iD2
1
2
3
13
B cis
OD OB i C B OD
Examples
3OB OA cis
iB2
3
2
1
iBD
iD2
1
2
3
1 3 3 1
2 2 2 2C i i
13
B cis
OD OB i C B OD
Examples
3OB OA cis
iB2
3
2
1
iBD
iD2
1
2
3
1 3 3 1
2 2 2 2C i i
iC
2
31
2
31
13
B cis
OD OB i C B OD
( ) i AP AO i
P A i O A
( ) i AP AO i
P A i O A
11 0 zizP
( ) i AP AO i
P A i O A
11 0 zizP
11 izzP
( ) i AP AO i
P A i O A
11 0 zizP
11 izzP
11 ziP
( ) i AP AO i
P A i O A
11 0 zizP
11 izzP
11 ziP
( ) i AP AO i ( ) ii QB BO i
P A i O A
11 0 zizP
11 izzP
11 ziP
Q B i O B
( ) i AP AO i ( ) ii QB BO i
P A i O A
11 0 zizP
11 izzP
11 ziP
Q B i O B
iBBQ
( ) i AP AO i ( ) ii QB BO i
P A i O A
11 0 zizP
11 izzP
11 ziP
Q B i O B
iBBQ
21 ziQ
( ) i AP AO i ( ) ii QB BO i
P A i O A
11 0 zizP
11 izzP
11 ziP
Q B i O B
iBBQ
21 ziQ
2
QPM
( ) i AP AO i ( ) ii QB BO i
P A i O A
11 0 zizP
11 izzP
11 ziP
Q B i O B
iBBQ
21 ziQ
2
QPM
2
11 21 ziziM
( ) i AP AO i ( ) ii QB BO i
P A i O A
11 0 zizP
11 izzP
11 ziP
Q B i O B
iBBQ
21 ziQ
2
QPM
2
11 21 ziziM
2
2121 izzzzM
( ) i AP AO i ( ) ii QB BO i
Exercise 4K; 1 to 8, 11, 12, 13
Exercise 4L; odd
Terry Lee: Exercise 2.6