x2 t01 09 geometrical representation of complex numbers
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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 DiagramNo 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 DiagramNo 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 InequalityIn 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 InequalityIn 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 InequalityIn 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 InequalityIn 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
AdditionIf a point A represents and point B represents then point C representing
is such that the points OACB form a parallelogram.
1z2z
21 zz
AdditionIf a point A represents and point B represents then point C representing
is such that the points OACB form a parallelogram.
1z2z
21 zz
SubtractionIf a point D represents and point E represents then the points ODEB form a parallelogram.
Note:
1z12 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 ThewOQ 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 ThewOQ 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 ThewOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangulatheUsing
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 ThewOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangulatheUsing
(ii) Construct the point R representing z + w, What can be said about thequadrilateral 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 ThewOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangulatheUsing
(ii) Construct the point R representing z + w, What can be said about thequadrilateral 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 ThewOQ is oflength The
wzPQ is oflength Thewzwz
OPQ
on inequalityr triangulatheUsing
(ii) Construct the point R representing z + w, What can be said about thequadrilateral OPRQ?
R
OPRQ is a parallelogram
?about said becan what , If zwwzwziii
?about said becan what , If zwwzwziii
wzwz
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
Multiplication
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
Multiplication
2121 zzzz
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
Multiplication
2121 zzzz 2121 argargarg zzzz
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
Multiplication
2121 zzzz 2121 argargarg zzzz
21212211 cisrrcisrcisr
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
Multiplication
2121 zzzz 2121 argargarg zzzz
21212211 cisrrcisrcisr
2
2121
by multiplied islength its and by iseanticlockwrotated isvector the,by multiply weif ..
rzzzei
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:
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
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 TAILy
x
A
O
C
B
)1(D
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 TAILy
x
A
O
C
B
)1(D
)1( iC
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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
( 1) ( 1)B i
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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
( 1) ( 1)B i
2by iseanticlockwrotation a is by tion Multiplica i
OR
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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
( 1) ( 1)B i
2by iseanticlockwrotation a is by tion Multiplica i
OR( 1) ( 1)B 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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
( 1) ( 1)B i
2by iseanticlockwrotation a is by tion Multiplica i
OR( 1) ( 1)B i
OR
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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
( 1) ( 1)B i
2by iseanticlockwrotation a is by tion Multiplica i
OR( 1) ( 1)B i
OR
2 ( 1)4
B cis
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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
( 1) ( 1)B i
2by iseanticlockwrotation a is by tion Multiplica i
OR( 1) ( 1)B i
OR
2 ( 1)4
B cis
1 ( 1)B 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
iC 2
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
iC 2
(ii) What complex number corresponds to the point of intersection D of the diagonals OB and 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
iC 2
(ii) What complex number corresponds to the point of intersection D of the diagonals OB and AC?
iiB21
2
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
iC 2
(ii) What complex number corresponds to the point of intersection D of the diagonals OB and AC?
iiB21
2
diagonals bisect in a rectangle
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
iC 2
(ii) What complex number corresponds to the point of intersection D of the diagonals OB and AC?
iiB21
2
diagonals bisect in a rectangle
iD 2121
Examples
Examples
31 cisB
Examples
31 cisB
iB23
21
Examples
31 cisB
iB23
21
iBD
Examples
31 cisB
iB23
21
iBD
iD21
23
Examples
31 cisB
iB23
21
iBD
iD21
23
DBC
Examples
31 cisB
iB23
21
iBD
iD21
23
DBC
iC
231
231
AOiAPi )(
AOiAPi )(
11 0 zizP
AOiAPi )(
11 0 zizP
11 izzP
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
2QPM
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
2QPM
2
11 21 ziziM
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
2QPM
2
11 21 ziziM
2
2121 izzzzM
Exercise 4K; 1 to 8, 11, 12, 13
Exercise 4L; odd