x2 t01 06 geometrical representation (2013)
DESCRIPTION
TRANSCRIPT
![Page 1: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/1.jpg)
Geometrical Representation of Complex Numbers
![Page 2: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/2.jpg)
Geometrical Representation of Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
![Page 3: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/3.jpg)
Geometrical Representation of Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.y
x
![Page 4: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/4.jpg)
Geometrical Representation of Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.y
x
iyxz
![Page 5: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/5.jpg)
Geometrical Representation of Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.y
x
iyxz
![Page 6: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/6.jpg)
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
![Page 7: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/7.jpg)
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
![Page 8: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/8.jpg)
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
![Page 9: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/9.jpg)
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
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Addition / Subtraction
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Addition / Subtraction
y
x
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Addition / Subtraction
y
x
1z
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Addition / Subtraction
y
x
1z
2z
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Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
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Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
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Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
![Page 17: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/17.jpg)
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)
![Page 18: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/18.jpg)
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
![Page 19: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/19.jpg)
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
![Page 20: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/20.jpg)
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
![Page 21: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/21.jpg)
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
![Page 22: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/22.jpg)
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
![Page 23: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/23.jpg)
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
![Page 24: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/24.jpg)
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)
![Page 25: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/25.jpg)
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
![Page 26: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/26.jpg)
AdditionIf a point A represents and point Brepresents then point C representing
is such that the points OACB form a parallelogram.
1z2z
21 zz
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AdditionIf a point A represents and point Brepresents 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
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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
![Page 29: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/29.jpg)
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
![Page 30: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/30.jpg)
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
![Page 31: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/31.jpg)
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
![Page 32: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/32.jpg)
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
![Page 33: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/33.jpg)
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
![Page 34: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/34.jpg)
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?
![Page 35: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/35.jpg)
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
![Page 36: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/36.jpg)
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
![Page 37: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/37.jpg)
?about said becan what , If zwwzwziii
![Page 38: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/38.jpg)
?about said becan what , If zwwzwziii
wzwz
![Page 39: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/39.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =
![Page 40: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/40.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
![Page 41: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/41.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
![Page 42: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/42.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw
![Page 43: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/43.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
![Page 44: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/44.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
2argarg
zw
2arg
zw imaginarypurely is
zw
Multiplication
![Page 45: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/45.jpg)
?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
![Page 46: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/46.jpg)
?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
![Page 47: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/47.jpg)
?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
![Page 48: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/48.jpg)
?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
![Page 49: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/49.jpg)
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
![Page 50: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/50.jpg)
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.
![Page 51: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/51.jpg)
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
![Page 52: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/52.jpg)
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
![Page 53: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/53.jpg)
y
x
A
O
C
B
)1(D
![Page 54: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/54.jpg)
y
x
A
O
C
B
)1(D
DC DA i
![Page 55: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/55.jpg)
y
x
A
O
C
B
)1(D
DC DA i
1 1C i
![Page 56: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/56.jpg)
y
x
A
O
C
B
)1(D
DC DA i
1 1C i
1 1
1
C i
i i
![Page 57: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/57.jpg)
y
x
A
O
C
B
)1(D
DC DA i
B A DC
1 1C i
1 1
1
C i
i i
![Page 58: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/58.jpg)
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
![Page 59: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/59.jpg)
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
![Page 60: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/60.jpg)
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
![Page 61: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/61.jpg)
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
![Page 62: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/62.jpg)
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
![Page 63: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/63.jpg)
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
![Page 64: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/64.jpg)
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
![Page 65: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/65.jpg)
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
![Page 66: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/66.jpg)
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
![Page 67: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/67.jpg)
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
![Page 68: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/68.jpg)
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
![Page 69: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/69.jpg)
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
![Page 70: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/70.jpg)
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
![Page 71: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/71.jpg)
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
![Page 72: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/72.jpg)
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
![Page 73: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/73.jpg)
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
![Page 74: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/74.jpg)
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
![Page 75: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/75.jpg)
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
![Page 76: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/76.jpg)
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
![Page 77: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/77.jpg)
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
2A CD
![Page 78: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/78.jpg)
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
2A CD
22
iD
![Page 79: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/79.jpg)
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
12
D i
2C i
midpoint of D AC
2A CD
22
iD
![Page 80: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/80.jpg)
Examples
![Page 81: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/81.jpg)
Examples
3OB OA cis
![Page 82: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/82.jpg)
Examples
3OB OA cis
13
B cis
![Page 83: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/83.jpg)
Examples
3OB OA cis
iB23
21
13
B cis
![Page 84: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/84.jpg)
Examples
3OB OA cis
iB23
21
13
B cis
OD OB i
![Page 85: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/85.jpg)
Examples
3OB OA cis
iB23
21
iBD
13
B cis
OD OB i
![Page 86: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/86.jpg)
Examples
3OB OA cis
iB23
21
iBD
iD21
23
13
B cis
OD OB i
![Page 87: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/87.jpg)
Examples
3OB OA cis
iB23
21
iBD
iD21
23
13
B cis
OD OB i C B OD
![Page 88: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/88.jpg)
Examples
3OB OA cis
iB23
21
iBD
iD21
23
1 3 3 12 2 2 2
C i i
13
B cis
OD OB i C B OD
![Page 89: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/89.jpg)
Examples
3OB OA cis
iB23
21
iBD
iD21
23
1 3 3 12 2 2 2
C i i
iC
231
231
13
B cis
OD OB i C B OD
![Page 90: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/90.jpg)
![Page 91: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/91.jpg)
( ) i AP AO i
![Page 92: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/92.jpg)
P A i O A
( ) i AP AO i
![Page 93: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/93.jpg)
P A i O A
11 0 zizP
( ) i AP AO i
![Page 94: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/94.jpg)
P A i O A
11 0 zizP
11 izzP
( ) i AP AO i
![Page 95: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/95.jpg)
P A i O A
11 0 zizP
11 izzP 11 ziP
( ) i AP AO i
![Page 96: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/96.jpg)
P A i O A
11 0 zizP
11 izzP 11 ziP
( ) i AP AO i
( ) ii QB BO i
![Page 97: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/97.jpg)
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
![Page 98: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/98.jpg)
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
![Page 99: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/99.jpg)
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
![Page 100: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/100.jpg)
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
![Page 101: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/101.jpg)
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
![Page 102: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/102.jpg)
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
![Page 103: X2 t01 06 geometrical representation (2013)](https://reader034.vdocuments.mx/reader034/viewer/2022042521/5486b509b47959e20c8b5307/html5/thumbnails/103.jpg)
Cambridge Ex 1E; 2 to 8, 10 to 15, 18, 19, 21, 22
Terry Lee: Exercise 2.6