X2 T01 09 geometrical representation of complex numbers
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Transcript of X2 T01 09 geometrical representation of complex numbers
![Page 1: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/1.jpg)
Geometrical Representation of Complex Numbers
![Page 2: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/2.jpg)
Geometrical Representation of Complex Numbers
Complex numbers can be represented on the Argand Diagram as vectors.
![Page 3: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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
![Page 10: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/10.jpg)
Addition / Subtraction
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Addition / Subtraction
y
x
![Page 12: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/12.jpg)
Addition / Subtraction
y
x
1z
![Page 13: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/13.jpg)
Addition / Subtraction
y
x
1z
2z
![Page 14: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/14.jpg)
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
![Page 15: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/15.jpg)
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
![Page 16: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/16.jpg)
Addition / Subtraction
y
x
1z
2z
To add two complex numbers, place the vectors “head to tail”
21 zz
![Page 17: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/26.jpg)
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
![Page 27: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/27.jpg)
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
![Page 28: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/28.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 29: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/37.jpg)
?about said becan what , If zwwzwziii
![Page 38: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/38.jpg)
?about said becan what , If zwwzwziii
wzwz
![Page 39: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/39.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =
![Page 40: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/40.jpg)
?about said becan what , If zwwzwziii
wzwz i.e. diagonals in OPRQ are =rectanglea is OPRQ
![Page 41: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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:
![Page 51: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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.
![Page 52: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/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.
2by iseanticlockwrotation a is by tion Multiplica i
![Page 53: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/53.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 54: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/54.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 TAILy
x
A
O
C
B
)1(D
2by iseanticlockwrotation a is by tion Multiplica i
![Page 55: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/55.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 TAILy
x
A
O
C
B
)1(D
)1( iC
2by iseanticlockwrotation a is by tion Multiplica i
![Page 56: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/56.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 TAILy
x
A
O
C
B
)1(D
)1( iC( 1) 1B i
2by iseanticlockwrotation a is by tion Multiplica i
![Page 57: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/57.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 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
![Page 58: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/58.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 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
![Page 59: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/59.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 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
![Page 60: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/60.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 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
![Page 61: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/61.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 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
![Page 62: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/62.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 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
![Page 63: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/63.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 64: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/64.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 65: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/65.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
iC 2
![Page 66: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/66.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
iC 2
(ii) What complex number corresponds to the point of intersection D of the diagonals OB and AC?
![Page 67: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/67.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
iC 2
(ii) What complex number corresponds to the point of intersection D of the diagonals OB and AC?
iiB21
2
![Page 68: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/68.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
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
![Page 69: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/69.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
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
![Page 70: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/70.jpg)
Examples
![Page 71: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/71.jpg)
Examples
31 cisB
![Page 72: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/72.jpg)
Examples
31 cisB
iB23
21
![Page 73: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/73.jpg)
Examples
31 cisB
iB23
21
iBD
![Page 74: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/74.jpg)
Examples
31 cisB
iB23
21
iBD
iD21
23
![Page 75: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/75.jpg)
Examples
31 cisB
iB23
21
iBD
iD21
23
DBC
![Page 76: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/76.jpg)
Examples
31 cisB
iB23
21
iBD
iD21
23
DBC
iC
231
231
![Page 77: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/77.jpg)
![Page 78: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/78.jpg)
AOiAPi )(
![Page 79: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/79.jpg)
AOiAPi )(
11 0 zizP
![Page 80: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/80.jpg)
AOiAPi )(
11 0 zizP
11 izzP
![Page 81: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/81.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
![Page 82: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/82.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(
![Page 83: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/83.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
![Page 84: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/84.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
![Page 85: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/85.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
2QPM
![Page 86: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/86.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
2QPM
2
11 21 ziziM
![Page 87: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/87.jpg)
AOiAPi )(
11 0 zizP
11 izzP 11 ziP
BOiBQii )(iBBQ
21 ziQ
2QPM
2
11 21 ziziM
2
2121 izzzzM
![Page 88: X2 T01 09 geometrical representation of complex numbers](https://reader034.fdocuments.in/reader034/viewer/2022052522/54825b0fb07959290c8b47a0/html5/thumbnails/88.jpg)
Exercise 4K; 1 to 8, 11, 12, 13
Exercise 4L; odd